DG-FEM for PDE’s Lecture 2Jan S Hesthaven Brown University [email protected] DG-FEM for PDE’s Lecture 2 Univ Coruna/USC/Univ Vigo 2010 Monday, June 7, 2010 A brief overview

Jan S HesthavenBrown [email protected]

DG-FEM for PDE’sLecture 2

Univ Coruna/USC/Univ Vigo 2010

Monday, June 7, 2010

mailto:[email protected]

mailto:[email protected]

A brief overview of what’s to come

‣ Lecture 1: Introduction and DG in 1D

‣ Lecture 2: Implementation and numerical aspects

‣ Lecture 3: Insight through theory

‣ Lecture 4: Nonlinear problems

‣ Lecture 5: Mesh generation and 2D DG-FEM

‣ Lecture 6: Implementation and applications

‣ Lecture 7: Higher order/Global problems

‣ Lecture 8: 3D and advanced topics


Let’s recall

We already know a lot about the basic DG-FEM

‣ Stability is provided by carefully choosing the numerical flux.‣ Accuracy appear to be given by the local solution representation.‣ We can utilize major advances on monotone schemes to design fluxes.‣ The scheme generalizes with very few changes to very general systems of conservation laws.

At least in principle -- but how we do it in practice ?


Recall the basic formulation

1 Introduction 9

of the scheme and is also where one can introduce knowledge of the dynam-ics of the problem [e.g., by upwinding through the flux as in a finite volumemethod (FVM)].

To mimic the terminology of the finite element scheme, we write the twolocal elementwise schemes as

Mk dukh

dt! (Sk)T fk

h !Mkgkh = !f!(xk+1)!k(xk+1) + f!(xk)!k(xk)

and

Mk dukh

dt+ Skfk

h !Mkgkh = (fk

h (xk+1) ! f!(xk+1))!k(xk+1)

!(fkh (xk) ! f!(xk))!k(xk). (1.5)

Here we have the vectors of local unknowns, ukh, of fluxes, fk

h, and the forc-ing, gk

h, all given on the nodes in each element. Given the duplication ofunknowns at the element interfaces, each vector is 2K long. Furthermore, wehave !k(x) = [!k

1(x), . . . , !kNp

(x)]T and the local matrices

Mkij =

!

Dk!ki (x)!k

j (x) dx, Skij =

!

Dk!ki (x)

d!kj

dxdx.

While the structure of the DG-FEM is very similar to that of the finite ele-ment method (FEM), there are several fundamental di!erences. In particular,the mass matrix is local rather than global and thus can be inverted at verylittle cost, yielding a semidiscrete scheme that is explicit. Furthermore, bycarefully designing the numerical flux to reflect the underlying dynamics, onehas more flexibility than in the classic FEM to ensure stability for wave-dominated problems. Compared with the FVM, the DG-FEM overcomes thekey limitation on achieving high-order accuracy on general grids by enablingthis through the local element-based basis. This is all achieved while main-taining benefits such as local conservation and flexibility in the choice of thenumerical flux.

All of this, however, comes at a price – most notably through an increasein the total degrees of freedom as a direct result of the decoupling of theelements. For linear elements, this yields a doubling in the total number ofdegrees of freedom compared to the continuous FEM as discussed above. Forcertain problems, this clearly becomes an issue of significant importance; thatis, if we seek a steady solution and need to invert the discrete operator Sin Eq. (1.5) then the associated computational work scales directly with thesize of the matrix. Furthermore, for problems where the flexibility in theflux choices and the locality of scheme is of less importance (e.g., for ellipticproblems), the DG-FEM is not as e"cient as a better suited method like theFEM.

On the other hand, for the DG-FEM, the operator tends to be more sparsethan for a FEM of similar order, potentially leading to a faster solution pro-cedure. This overhead, caused by the doubling of degrees of freedom along

Weak:

Strong:

We introduce the local notation

8 1 Introduction

Here we have the vectors of local unknowns, uk, of fluxes, fk, and the forcing, gk, all given on thenodes in each element. Given the duplication of unknowns at the element interfaces, each vector is2K long. Furthermore, we have k(x) = [k

1(x), . . . , kNp


Mkij =

Dkki (x)k

j (x) dx, Skij =

Dkki (x)

dkj

dxdx.

While the structure of the DG-FEM is very similar to that of the finite element method (FEM), thereare several fundamental differences. In particular, the mass matrix is local rather than global andthus can be inverted at very little cost, yielding a semidiscrete scheme that is explicit. Furthermore,by carefully designing the numerical flux to reflect the underlying dynamics, one has more flexibilitythan in the classic FEM to ensure stability for wave-dominated problems. Compared with the FVM,the DG-FEM overcomes the key limitation on achieving high-order accuracy on general grids byenabling this through the local element-based basis. This is all achieved while maintaining benefitssuch as local conservation and flexibility in the choice of the numerical flux.

All of this, however, comes at a price – most notably through an increase in the total degreesof freedom as a direct result of the decoupling of the elements. For linear elements, this yields adoubling in the total number of degrees of freedom compared to the continuous FEM as discussedabove. For certain problems, this clearly becomes an issue of significant importance; that is, if weseek a steady solution and need to invert the discrete operator S in Eq.(1.5) then the associatedcomputational work scales directly with the size of the matrix. Furthermore, for problems wherethe flexibility in the flux choices and the locality of scheme is of less importance (e.g., for ellipticproblems), the DG-FEM is not as efficient as a better suited method like the FEM.

On the other hand, for the DG-FEM, the operator tends to be more sparse than for a FEMof similar order, potentially leading to a faster solution procedure. This overhead, caused by thedoubling of degrees of freedom along element interfaces, decreases as the order of elements increase,thus reducing the penalty of the doubling of the interface nodes somewhat. In Table 1.1 we alsosummarize the strengths and weaknesses of the DG-FEM. This simple – and simpleminded – com-parative discussion also highlights that time dependent wave-dominated problems emerge as themain candidates for problems where DG-FEM will be advantageous. As we will see shortly, this isalso where they have achieved most prominence and widespread use during the last decade.

1.1 A brief account of history

The discontinuous Galerkin finite element method (DG-FEM) appears to have been proposed firstin [269] as a way of solving the steady-state neutron transport equation

σu +∇ · (au) = f,

where σ is a real constant, a(x) is piecewise constant, and u is the unknown. The first analysis ofthis method was presented in [217], showing O(hN )-convergence on a general triangulated grid andoptimal convergence rate, O(hN+1), on a Cartesian grid of cell size h and with a local polynomialapproximation of order N . This result was later improved in [190] to O(hN+1/2)-convergence ongeneral grids. The optimality of this convergence rate was subsequently confirmed in [257] by a spe-cial example. These results assume smooth solutions, whereas the linear problems with nonsmoothsolutions were analyzed in [63, 225]. Techniques for postprocessing on Cartesian grids to enhance

to obtain the semi-discrete forms of DG-FEM

The basics of DG-FEM

1 Introduction 9



Mk dukh

dt! (Sk)T fk


and

Mk dukh

dt+ Skfk

h !Mkgkh = (fk

h (xk+1) ! f!(xk+1))!k(xk+1)

!(fkh (xk) ! f!(xk))!k(xk). (1.5)




1(x), . . . , !kNp


Mkij =

!

Dk!ki (x)!k

j (x) dx, Skij =

!

Dk!ki (x)

d!kj

dxdx.




1 Introduction 9



Mk dukh

dt! (Sk)T fk


and

Mk dukh

dt+ Skfk

h !Mkgkh = (fk

h (xk+1) ! f!(xk+1))!k(xk+1)

!(fkh (xk) ! f!(xk))!k(xk). (1.5)




1(x), . . . , !kNp


Mkij =

!

Dk!ki (x)!k

j (x) dx, Skij =

!

Dk!ki (x)

d!kj

dxdx.




Weak:

Strong:

To simplify the notation, introduce

8 1 Introduction


1(x), . . . , kNp


Mkij =

Dkki (x)k

j (x) dx, Skij =

Dkki (x)

dkj

dxdx.






σu +∇ · (au) = f,


to obtain the two basic forms of DG-FEM


.. but how do we bring this to life ?

We should seek

‣ A flexible and generic framework for solving 1D problems using DG-FEM

‣ Easy maintenance and setup

‣ Separation of grids, discretization, and physics

‣ Easy implementation

Requirements

What do we want?

! A flexible and generic framework useful for solving di!erentproblems

! Easy maintenance by a component-based setup

! Splitting of the mesh and solver problems for reusability

! Easy implementation

Let’s see how this can be achieved...

3 / 58

Let’s see if we can achieve thisMonday, June 7, 2010

The local approximation

We recall that we have

where the computational domain is

Domain of interest

We want to solve a given problem in a domain ! which isapproximated by the union of K nonoverlapping local elements Dk ,k = 1, ..., K such that

! != !h =K!

k=1

Dk

Thus, we need to deal with implementation issues with respect tothe local elements and how they are related.

The shape of the local elements can in principle be of any shape,however, in practice we mostly consider d-dimensional simplexes(e.g. triangles in two dimensions).

4 / 58

Sketch and notations for a one-dimensional domain

Consider a one-dimensional domain defined on x ! [L, R]

xDk!1 Dk Dk+1

hk+1

L = x1l xk!1

r = xkl xk

r = xk+1l xK

r = R

5 / 58



3

Making it work in one dimension

As simple as the formulations in the last chapter appear, there is often aleap between mathematical formulations and an actual implementation of thealgorithms. This is particularly true when one considers important issues suchas e!ciency, flexibility, and robustness of the resulting methods.

In this chapter we address these issues by first discussing details such asthe form of the local basis and, subsequently, how one implements the nodalDG-FEMs in a flexible way. To keep things simple, we continue the emphasison one-dimensional linear problems, although this results in a few apparentlyunnecessarily complex constructions. We ask the reader to bear with us, asthis slightly more general approach will pay o" when we begin to considermore complex nonlinear and/or higher-dimensional problems.

3.1 Legendre polynomials and nodal elements

In Chapter 2 we started out by assuming that one can represent the globalsolution as a the direct sum of local piecewise polynomial solution as

u(x, t) ! uh(x, t) =K!

k=1

ukh(xk, t).

The careful reader will note that this notation is a bit careless, as we do notaddress what exactly happens at the overlapping interfaces. However, a morecareful definition does not add anything essential at this point and we will usethis notation to reflect that the global solution is obtained by combining theK local solutions as defined by the scheme.

The local solutions are assumed to be of the form

x " Dk = [xkl , xk

r ] : ukh(x, t) =

Np"

n=1

ukn(t)!n(x) =

Np"

i=1

ukh(xk

i , t)"ki (x).

On each element we have

using either a modal or a nodal representation

We introduce an affine mapping as

Local approximation in 1D

On each of the local elements, we choose to represent the solutionlocally as a polynomial of arbitrary order N = Np ! 1 as

x " Dk : ukh (x(r), t) =

Np!

n=1

ukn (t)!n(r) =

Np!

i=1

uki (t)lki (r)

using either a modal or nodal representation on r " I ([!1, 1]).

We have introduced the a!ne mapping from the standard elementI to the k ’th element Dk

x " Dk : x(r) = xkl +

1 + r

2hk , hk = xk

r ! xkl

Dk

xkl xk

r

I

!1 r 1

6 / 58Monday, June 7, 2010


Modal representation is based on Legendre polynomials


The modal representation is usually based on a hierachical modalbasis, e.g. the normalized Legendre polynomials from the class oforthogonal Jacobi polynomials

!n(r) = Pn!1(r) =Pn!1(r)!

"n!1, "n =

2

2n + 1

Note, the orthogonal property implies that for a normalized basis

! 1

!1Pi (r)Pj(r)dr = #ij

The nodal representation is based on the interpolating Lagrangepolynomials, which enjoys the Cardinal property

lki (x) = #ij , #ij =

"

0 , i "= j

1 , i = j

7 / 58



!n(r) = Pn!1(r) =Pn!1(r)!

"n!1, "n =

2

2n + 1


! 1



lki (x) = #ij , #ij =

"

0 , i "= j

1 , i = j

7 / 58

Local approximation in 1D - modesInterpolating normalized Legendre polynomials !n(r) = Pn!1(r).

−1 0 1−202 !1(r)

−1 0 1−202 !2(r)

−1 0 1−202 !3(r)

−1 0 1−202 !4(r)

−1 0 1−202 !5(r)

−1 0 1−202 !6(r)

8 / 58



Nodal representation based on Lagrange polynomials



!n(r) = Pn!1(r) =Pn!1(r)!

"n!1, "n =

2

2n + 1


! 1



lki (x) = #ij , #ij =

"

0 , i "= j

1 , i = j

7 / 58

Local approximation in 1D - nodesInterpolating Lagrange polynomials ln(r) based on theLegendre-Gauss-Lobatto (LGL) nodes.

−1 0 1−1012

l1(r)

−1 0 1−1012

l2(r)

−1 0 1−1012

l3(r)

−1 0 1−1012

l4(r)

−1 0 1−1012

l5(r)

−1 0 1−1012

l6(r)

9 / 58Monday, June 7, 2010

Local approximation in 1DFor stable and accurate computations we need to ensure that thegeneralized Vandermonde matrix V is well-conditioned. Thisimplies minimizing the Lebesque constant

! = maxr

Np!

i=1

|li (r)|

and maximizing Det V.

0 10 20 30 4010−10

100

1010

1020

1030

Det

V

N0 10 20 30 40

100

102

104

106

108

1010

!(V)

N

EquiNormalized LGL

11 / 58


These two representations are connected as


2

The key ideas

2.1 Briefly on notation

While we initially strive to stay away from complex notation a few funda-mentals are needed. We consider problems posed on the physical domain !with boundary "! and assume that this domain is well approximated bythe computational domain !h. This is a space filling triangulation composedof a collection of K geometry-conforming nonoverlapping elements, Dk. Theshape of these elements can be arbitrary although we will mostly considercases where they are d-dimensional simplexes.

We define the local inner product and L2(Dk) norm

(u, v)Dk =!

Dkuv dx, !u!2

Dk = (u, u)Dk ,

as well as the global broken inner product and norm

(u, v)!,h =K"

k=1

(u, v)Dk , !u!2!,h = (u, u)!,h .

Here, (!, h) reflects that ! is only approximated by the union of Dk, that is

! " !h =K#

k=1

Dk,

although we will not distinguish the two domains unless needed.Generally, one has local information as well as information from the neigh-

boring element along an intersection between two elements. Often we will referto the union of these intersections in an element as the trace of the element.For the methods we discuss here, we will have two or more solutions or bound-ary conditions at the same physical location along the trace of the element.

16 2 The key ideas

where u can be both a scalar and a vector. In a similar fashion we also define the jumps along anormal, n, as

[[u]] = n−u− + n+u+, [[u]] = n− · u− + n+ · u+.

Note that it is defined differently depending on whether u is a scalar or a vector, u.

2.2 Basic elements of the schemes

In the following, we introduce the key ideas behind the family of discontinuous element methodsthat are the main topic of this text. Before getting into generalizations and abstract ideas, let usdevelop a basic understanding of the schemes through a simple example.

2.2.1 The first schemes

Consider the linear scalar wave equation

∂u

∂t+

∂f(u)∂x

= 0, x ∈ [L,R] = Ω, (2.1)

where the linear flux is given as f(u) = au. This is subject to the appropriate initial conditions

u(x, 0) = u0(x).

Boundary conditions are given when the boundary is an inflow boundary, that is

u(L, t) = g(t) if a ≥ 0,u(R, t) = g(t) if a ≤ 0.

We approximate Ω by K nonoverlapping elements, x ∈ [xkl , xk

r ] = Dk, as illustrated in Fig. 2.1. Oneach of these elements we express the local solution as a polynomial of order N

x ∈ Dk : ukh(x, t) =

Np

n=1

ukn(t)ψn(x) =

Np

i=1

ukh(xk

i , t)ki (x).

Here, we have introduced two complementary expressions for the local solution. In the first one,known as the modal form, we use a local polynomial basis, ψn(x). A simple example of this couldbe ψn(x) = xn−1. In the alternative form, known as the nodal representation, we introduce Np =N + 1 local grid points, xk

i ∈ Dk, and express the polynomial through the associated interpolatingLagrange polynomial, k

i (x). The connection between these two forms is through the definition ofthe expansion coefficients, uk

n. We return to a discussion of these choices in much more detail later;for now it suffices to assume that we have chosen one of these representations.

The global solution u(x, t) is then assumed to be approximated by the piecewise N -th orderpolynomial approximation uh(x, t),

u(x, t) uh(x, t) =K

k=1

ukh(x, t),

3






u(x, t) ! uh(x, t) =K!

k=1

ukh(xk, t).



x " Dk = [xkl , xk

r ] : ukh(x, t) =

Np"

n=1

ukn(t)!n(x) =

Np"

i=1

ukh(xk

i , t)"ki (x).

50 3 Making it work in one dimension

!1.25 !1 !0.75 !0.5 !0.25 0 0.25!1

0123456789

10

!i

"

0 2 4 6 8 10100

105

1010

1015

1020

1025

"

Det

erm

inan

t of V

N=4N=8

N=16

N=24

N=32

Fig. 3.3. On the left is shown the location of the left half of the Jacobi-Gauss-Lobatto nodes for N = 16 as a function of !, the type of the polynomial. Note that! = 0 corresponds to the LGL nodes. On the right is shown the behavior of theVandermonde determinant as a function of ! for di!erent values of N , confirmingthe optimality of the LGL nodes but also showing robustness of the interpolationto this choice.

decay in the value of the determinant, indicating possible problems in theinterpolation. Considering the corresponding nodal distribution in Fig. 3.3for these values of !, this is perhaps not surprising.

This highlights that it is the overall structure of the nodes rather thanthe details of the individual node position that is important; for example, onecould optimize these nodal sets for various applications.

To summarize matters, we have local approximations of the form

u(r) ! uh(r) =Np!

n=1

unPn!1(r) =Np!

i=1

u(ri)"i(r), (3.3)

where #i = ri are the Legendre-Gauss-Lobatto quadrature points. A centralcomponent of this construction is the Vandermonde matrix, V, which estab-lishes the connections

u = Vu, VT !(r) = P (r), Vij = Pj(ri).

By carefully choosing the orthonormal Legendre basis, Pn(r), and the nodalpoints, ri, we have ensured that V is a well-conditioned object and that theresulting interpolation is well behaved. A script for initializing V is given inVandermonde1D.m.


!1.25 !1 !0.75 !0.5 !0.25 0 0.25!1

0123456789

10

!i

"

0 2 4 6 8 10100

105

1010

1015

1020

1025

"

Det

erm

inan

t of V

N=4N=8

N=16

N=24

N=32





u(r) ! uh(r) =Np!

n=1

unPn!1(r) =Np!

i=1

u(ri)"i(r), (3.3)





We did not, however, discuss the specifics of this representation, as this is lessimportant from a theoretical point of view. The results in Example 2.4 clearlyillustrate, however, that the accuracy of the method is closely linked to theorder of the local polynomial representation and some care is warranted whenchoosing this.

Let us begin by introducing the a!ne mapping

x ! Dk : x(r) = xkl +

1 + r

2hk, hk = xk

r " xkl , (3.1)

with the reference variable r ! I = ["1, 1]. We consider local polynomialrepresentations of the form

x ! Dk : ukh(x(r), t) =

Np!

n=1

ukn(t)!n(r) =

Np!

i=1

ukh(xk

i , t)"ki (r).

Let us first discuss the local modal expansion,

uh(r) =Np!

n=1

un!n(r).

where we have dropped the superscripts for element k and the explicit timedependence, t, for clarity of notation.

As a first choice, one could consider !n(r) = rn!1 (i.e., the simple mono-mial basis). This leaves only the question of how to recover un. A natural wayis by an L2-projection; that is by requiring that

(u(r),!m(r))I =Np!

n=1

un (!n(r),!m(r))I ,

for each of the Np basis functions !n. We have introduced the inner producton the interval I as

(u, v)I =" 1

!1uv dx.

This yieldsMu = u,

whereMij = (!i,!j)I , u = [u1, . . . , uNp ]T , ui = (u,!i)I ,

leading to Np equations for the Np unknown expansion coe!cients, ui. How-ever, note that

Mij =1

i + j " 1#1 + ("1)i+j

$, (3.2)

which resembles a Hilbert matrix, known to be very poorly conditioned. If wecompute the condition number, #(M), for M for increasing order of approx-imation, N , we observe in Table 3.1 the very rapidly deteriorating condition-ing of M. The reason for this is evident in Eq. (3.2), where the coe!cient

r ∈ [−1, 1]


To make this accurate, we have made some choices


2

The key ideas

2.1 Briefly on notation

While we initially strive to stay away from complex notation a few funda-mentals are needed. We consider problems posed on the physical domain !with boundary "! and assume that this domain is well approximated bythe computational domain !h. This is a space filling triangulation composedof a collection of K geometry-conforming nonoverlapping elements, Dk. Theshape of these elements can be arbitrary although we will mostly considercases where they are d-dimensional simplexes.

We define the local inner product and L2(Dk) norm

(u, v)Dk =!

Dkuv dx, !u!2

Dk = (u, u)Dk ,

as well as the global broken inner product and norm

(u, v)!,h =K"

k=1

(u, v)Dk , !u!2!,h = (u, u)!,h .

Here, (!, h) reflects that ! is only approximated by the union of Dk, that is

! " !h =K#

k=1

Dk,

although we will not distinguish the two domains unless needed.Generally, one has local information as well as information from the neigh-

boring element along an intersection between two elements. Often we will referto the union of these intersections in an element as the trace of the element.For the methods we discuss here, we will have two or more solutions or bound-ary conditions at the same physical location along the trace of the element.

16 2 The key ideas

where u can be both a scalar and a vector. In a similar fashion we also define the jumps along anormal, n, as

[[u]] = n−u− + n+u+, [[u]] = n− · u− + n+ · u+.

Note that it is defined differently depending on whether u is a scalar or a vector, u.

2.2 Basic elements of the schemes

In the following, we introduce the key ideas behind the family of discontinuous element methodsthat are the main topic of this text. Before getting into generalizations and abstract ideas, let usdevelop a basic understanding of the schemes through a simple example.

2.2.1 The first schemes

Consider the linear scalar wave equation

∂u

∂t+

∂f(u)∂x

= 0, x ∈ [L,R] = Ω, (2.1)

where the linear flux is given as f(u) = au. This is subject to the appropriate initial conditions

u(x, 0) = u0(x).

Boundary conditions are given when the boundary is an inflow boundary, that is

u(L, t) = g(t) if a ≥ 0,u(R, t) = g(t) if a ≤ 0.

We approximate Ω by K nonoverlapping elements, x ∈ [xkl , xk

r ] = Dk, as illustrated in Fig. 2.1. Oneach of these elements we express the local solution as a polynomial of order N

x ∈ Dk : ukh(x, t) =

Np

n=1

ukn(t)ψn(x) =

Np

i=1

ukh(xk

i , t)ki (x).

Here, we have introduced two complementary expressions for the local solution. In the first one,known as the modal form, we use a local polynomial basis, ψn(x). A simple example of this couldbe ψn(x) = xn−1. In the alternative form, known as the nodal representation, we introduce Np =N + 1 local grid points, xk

i ∈ Dk, and express the polynomial through the associated interpolatingLagrange polynomial, k

i (x). The connection between these two forms is through the definition ofthe expansion coefficients, uk

n. We return to a discussion of these choices in much more detail later;for now it suffices to assume that we have chosen one of these representations.

The global solution u(x, t) is then assumed to be approximated by the piecewise N -th orderpolynomial approximation uh(x, t),

u(x, t) uh(x, t) =K

k=1

ukh(x, t),

3






u(x, t) ! uh(x, t) =K!

k=1

ukh(xk, t).



x " Dk = [xkl , xk

r ] : ukh(x, t) =

Np"

n=1

ukn(t)!n(x) =

Np"

i=1

ukh(xk

i , t)"ki (x).


!1.25 !1 !0.75 !0.5 !0.25 0 0.25!1

0123456789

10

!i

"

0 2 4 6 8 10100

105

1010

1015

1020

1025

"

Det

erm

inan

t of V

N=4N=8

N=16

N=24

N=32





u(r) ! uh(r) =Np!

n=1

unPn!1(r) =Np!

i=1

u(ri)"i(r), (3.3)





!1.25 !1 !0.75 !0.5 !0.25 0 0.25!1

0123456789

10

!i

"

0 2 4 6 8 10100

105

1010

1015

1020

1025

"

Det

erm

inan

t of V

N=4N=8

N=16

N=24

N=32





u(r) ! uh(r) =Np!

n=1

unPn!1(r) =Np!

i=1

u(ri)"i(r), (3.3)





We did not, however, discuss the specifics of this representation, as this is lessimportant from a theoretical point of view. The results in Example 2.4 clearlyillustrate, however, that the accuracy of the method is closely linked to theorder of the local polynomial representation and some care is warranted whenchoosing this.

Let us begin by introducing the a!ne mapping

x ! Dk : x(r) = xkl +

1 + r

2hk, hk = xk

r " xkl , (3.1)

with the reference variable r ! I = ["1, 1]. We consider local polynomialrepresentations of the form

x ! Dk : ukh(x(r), t) =

Np!

n=1

ukn(t)!n(r) =

Np!

i=1

ukh(xk

i , t)"ki (r).

Let us first discuss the local modal expansion,

uh(r) =Np!

n=1

un!n(r).

where we have dropped the superscripts for element k and the explicit timedependence, t, for clarity of notation.

As a first choice, one could consider !n(r) = rn!1 (i.e., the simple mono-mial basis). This leaves only the question of how to recover un. A natural wayis by an L2-projection; that is by requiring that

(u(r),!m(r))I =Np!

n=1

un (!n(r),!m(r))I ,

for each of the Np basis functions !n. We have introduced the inner producton the interval I as

(u, v)I =" 1

!1uv dx.

This yieldsMu = u,

whereMij = (!i,!j)I , u = [u1, . . . , uNp ]T , ui = (u,!i)I ,

leading to Np equations for the Np unknown expansion coe!cients, ui. How-ever, note that

Mij =1

i + j " 1#1 + ("1)i+j

$, (3.2)

which resembles a Hilbert matrix, known to be very poorly conditioned. If wecompute the condition number, #(M), for M for increasing order of approx-imation, N , we observe in Table 3.1 the very rapidly deteriorating condition-ing of M. The reason for this is evident in Eq. (3.2), where the coe!cient

r ∈ [−1, 1]


To make this accurate, we have made some choices

Conditioning of V is determined by choice of points

Optimal points: Legendre Gauss Lobatto points

Important for robustness: Orthogonal local basis and

good points for interpolation



To implement a typical semi-discrete scheme

8 1 Introduction


1(x), . . . , kNp


Mkij =

Dkki (x)k

j (x) dx, Skij =

Dkki (x)

dkj

dxdx.






σu +∇ · (au) = f,


We need several local operators

Consider first the local mass-matrix

3.2 Elementwise operations 51

Vandermonde1D.m

function [V1D] = Vandermonde1D(N,r)

% function [V1D] = Vandermonde1D(N,r)% Purpose : Initialize the 1D Vandermonde Matrix, V_ij = phi_j(r_i);

V1D = zeros(length(r),N+1);for j=1:N+1

V1D(:,j) = JacobiP(r(:), 0, 0, j-1);end;return

Using V, we can transform directly between modal representations, usingun as the unknowns, and nodal representations, using u(ri) as the unknowns.The two representations are mathematically equivalent but computationallydi!erent and they each have certain advantages and disadvantages, dependingon the application at hand.

In what remains, we will focus on the nodal representations, which hasadvantages for more complex problems as we will see later. However, it isimportant to keep the alternative in mind as an optional representation.

3.2 Elementwise operations

With the development of a suitable local representation of the solution, weare now prepared to discuss the various elementwise operators needed to im-plement the algorithms from Chapter 2.

We begin with the two operators Mk and Sk. For the former, the entriesare given as

Mkij =

! xkr

xkl

!ki (x)!k

j (x) dx =hk

2

! 1

!1!i(r)!j(r) dr =

hk

2(!i, !j)I =

hk

2Mij ,

where the coe"cient in front is the Jacobian coming from Eq. (3.1) and M isthe mass matrix defined on the reference element I.

Recall that

!i(r) =Np"

n=1

(VT )!1in Pn!1(r),

from which we recover

Mij =! 1

!1

Np"

n=1

(VT )!1in Pn!1(r)

Np"

m=1

(VT )!1jmPm!1(r) dr

=Np"

n=1

Np"

m=1

(VT )!1in (VT )!1

jm(Pn!1, Pm!1)I =Np"

n=1

(VT )!1in (VT )!1

jn ,

20 2 The key ideas

where boundary conditions are needed. Clearly, it is reasonable that the numerical approximation

to the wave equation displays similar behavior.

Assume that we use a nodal approach, i.e.,

x ∈ Dk: uk

h(x, t) =

Np

i=1

ukh(xk

i , t)ki (x),

and consider the strong form given as

Mk d

dtuk

h + Skauk

h

=

k

(x)(aukh − (auh)

∗)

xkr

xkl

. (2.8)

Here, we have introduced the nodal versions of the local operators

Mkij =

ki , k

j

Dk , Sk

ij =

ki ,

dkj

dx

Dk

.

First, realize that

uThMkuh =

Dk

Np

i=1

ukh(xk

i )ki (x)

Np

j=1

ukh(xk

j )kj (x) dx =

ukh

2

Dk ;

that is, it recovers the local energy. Furthermore, consider

uThSkuh =

Dk

Np

i=1

ukh(xk

i )ki (x)

Np

j=1

ukh(xk

j )dk

j

dxdx =

Dkuk

h(x)(ukh(x))

dx =1

2[(uk

h)2]xk

r

xkl.

Thus, it mimics an integration-by-parts form.

By multiplying Eq. (2.8) with uTh we directly recover

d

dt

ukh

2

Dk = −a[(ukh)

2]xk

r

xkl

+ 2uk

h(aukh − (au)

∗)xk

r

xkl. (2.9)

For stability, we must, as for the original equation, require that

K

k=1

d

dt

ukh

2

Dk =d

dtuh2Ω,h ≤ 0.

It suffices to control the terms associated with the coupling of the interfaces, each of which looks

like

n− ·au2

h(x−)− 2uh(x−)(au)∗(x−)

+ n+ ·

au2

h(x+)− 2uh(x+

)(au)∗(x+

)≤ 0

at every interface. Here (x−, x+) refers to the left (e.g., xk

r ), and right side (e.g., xk+1l ) of the

interface, respectively. We also have that n−= −n+

.

Let us consider a numerical flux like

f∗ = (au)∗

= au+ |a|1− α

2[[u]].



Recall first

Element-wise operationsEarlier, a relationship between the coe!cients of our modal andnodal expansions was found

u = Vu

Also, we can relate the modal and nodal bases through

uT l(r) = uT!(r)

These relationships can be combined to relate the basis functionsthrough V

uT l(r) = uT!(r)

! (Vu)T l(r) = uT!(r)

!uTVT l(r) = uT!(r)

!VT l(r) = !(r)

17 / 58

We also have


u = Vu


uT l(r) = uT!(r)


uT l(r) = uT!(r)

! (Vu)T l(r) = uT!(r)

!uTVT l(r) = uT!(r)

!VT l(r) = !(r)

17 / 58

This directly yields


u = Vu


uT l(r) = uT!(r)


uT l(r) = uT!(r)

! (Vu)T l(r) = uT!(r)

!uTVT l(r) = uT!(r)

!VT l(r) = !(r)

17 / 58



Using this, we recover


Vandermonde1D.m










Mkij =

! xkr

xkl

!ki (x)!k

j (x) dx =hk

2

! 1

!1!i(r)!j(r) dr =

hk

2(!i, !j)I =

hk

2Mij ,


Recall that

!i(r) =Np"

n=1

(VT )!1in Pn!1(r),


Mij =! 1

!1

Np"

n=1

(VT )!1in Pn!1(r)

Np"

m=1

(VT )!1jmPm!1(r) dr

=Np"

n=1

Np"

m=1

(VT )!1in (VT )!1


n=1

(VT )!1in (VT )!1

jn ,52 3 Making it work in one dimension

where the latter follows from orthonormality of Pn(r). Thus, we recover

Mk =hk

2M =

hk

2!VVT

"!1.

Let us now consider Sk, given as

Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .

Note in particular that no metric constant is introduced by the transformation.To realize a simple way to compute this, we define the di!erentiation matrix,Dr, with the entries

Dr,(i,j) =d!j

dr

$$$$ri

.

The motivation for this is found in Eq. (3.3), where we observe that Dr isthe operator that transforms point values, u(ri), to derivatives at these samepoints (e.g., u"

h = Druh).Consider now the product MDr, with entries

(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .

In other words, we have the identity

MDr = S.

The entries of the di!erentiation matrix, Dr, can be found directly,

VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to

VTDTr = (Vr)T , Vr,(i,j) =

dPj

dr

$$$$$ri

.

To compute the entries of Vr we use the identity

dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),

where P (1,1)n!1 is the Jacobi polynomial (see Appendix A). This is implemented

in GradJacobiP.m to initialize the gradient of V evaluated at the grid points,r, as in GradVandermonde1D.m.

Consider now the local stiffness matrix


Using this, we recover


Vandermonde1D.m










Mkij =

! xkr

xkl

!ki (x)!k

j (x) dx =hk

2

! 1

!1!i(r)!j(r) dr =

hk

2(!i, !j)I =

hk

2Mij ,


Recall that

!i(r) =Np"

n=1

(VT )!1in Pn!1(r),


Mij =! 1

!1

Np"

n=1

(VT )!1in Pn!1(r)

Np"

m=1

(VT )!1jmPm!1(r) dr

=Np"

n=1

Np"

m=1

(VT )!1in (VT )!1


n=1

(VT )!1in (VT )!1

jn ,52 3 Making it work in one dimension


Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),



Consider now the local stiffness matrix



Define first the nodal differentiation matrix

Consider



Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),




Define first the nodal differentiation matrix



Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),



Consider



Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





Mk =hk

2M =

hk

2!VVT

"!1.


Skij =

# xkr

xkl

!ki (x)

d!kj (x)dx

dx =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


Dr,(i,j) =d!j

dr

$$$$ri

.



(MDr)(i,j) =Np%

n=1

MinDr,(n,j) =Np%

n=1

# 1

!1!i(r)!n(r)

d!j

dr

$$$$rn

dr

=# 1

!1!i(r)

Np%

n=1

d!j

dr

$$$$rn

!n(r) dr =# 1

!1!i(r)

d!j(r)dr

dr = Sij .


MDr = S.


VT !(r) = P (r) ! VT d

dr!(r) =

d

drP (r),

leading to


dPj

dr

$$$$$ri

.


dPn

dr=

&n(n + 1)P (1,1)

n!1 (r),





A list of essential functions to do local operations are

Element-wise operations

A summary of useful scripts for the 1D element operations inMatlab

JacobiP Evaluate normalized Jacobi polynomialGradJacobiP Evaluate derivative of Jacobi polynomialJacobiGQ Compute Gauss points and weights for use in quadratureJacobiGL Compute the Legende-Gauss-Lobatto (LGL) nodesVandermonde1D Compute VGradVandermonde1D Compute Vr

Dmatrix1D Compute Dr

23 / 58

These are entirely problem independent


! 1

!1n · (uh ! u")!i(r) dr = (uh ! u")|rNp

eNp ! (uh ! u")|r1e1,

where ei is an Np long zero vector with a value of 1 in position i. Thus, it willbe convenient to define an operator that mimics this, as initialized in Lift1D.m,which returns the matrix, M!1E , where E is an Np " 2 array. Note that wehave again multiplied by M!1 for reasons that will become clear shortly.

Lift1D.m

function [LIFT] = Lift1D

% function [LIFT] = Lift1D% Purpose : Compute surface integral term in DG formulation

Globals1D;Emat = zeros(Np,Nfaces*Nfp);

% Define EmatEmat(1,1) = 1.0; Emat(Np,2) = 1.0;

% inv(mass matrix)*\s_n (L_i,L_j)_edge_nLIFT = V*(V’*Emat);return

3.3 Getting the grid together and computing the metric

With the local operators in place, we now discuss how to compute the metricassociated with the grid and how to assemble the global grid. We begin byassuming that we provide two sets of data:

• A row vector, V X, of Nv (= K + 1) vertex coordinates. These representthe end points of the intervals or elements. They do not need to be orderedin any particular way although they must be distinct. These vertices arenumbered consecutively from 1 to Nv.

• An integer matrix, EToV, of size K " 2, containing in each row k, thenumbers of the two vertices forming element k. It is the responsibility ofthe user/grid generator to ensure that the elements are meaningful; that is,the grid should consist of one connected set of nonoverlapping elements. Itis furthermore assumed that the nodes are ordered so elements are definedcounter-clockwise (in one dimension this reflects a left-to-right ordering ofthe elements).

These two pieces of information are provided by any grid generator, or,alternatively, a user should provide them to specify the grid for a problem(see Appendix B for a further discussion of this).

We will also need a boundary operator like


The local approximation - example

Consider


!!0.50 0.50!0.50 0.50

",

#!1.50 2.00 !0.50!0.50 0.00 0.50

0.50 !2.00 1.50

$,

%

&&&'

!5.00 6.76 !2.67 1.41 !0.50!1.24 0.00 1.75 !0.76 0.26

0.38 !1.34 0.00 1.34 !0.38!0.26 0.76 !1.75 0.00 1.24

0.50 !1.41 2.67 !6.76 5.00

(

)))*,

%

&&&&&&&&&&'

!18.00 24.35 !9.75 5.54 !3.66 2.59 !1.87 1.28 !0.50!4.09 0.00 5.79 !2.70 1.67 !1.15 0.82 !0.56 0.22

0.99 !3.49 0.00 3.58 !1.72 1.08 !0.74 0.49 !0.19!0.44 1.29 !2.83 0.00 2.85 !1.38 0.86 !0.55 0.21

0.27 !0.74 1.27 !2.66 0.00 2.66 !1.27 0.74 !0.27!0.21 0.55 !0.86 1.38 !2.85 0.00 2.83 !1.29 0.44

0.19 !0.49 0.74 !1.08 1.72 !3.58 0.00 3.49 !0.99!0.22 0.56 !0.82 1.15 !1.67 2.70 !5.79 0.00 4.09

0.50 !1.28 1.87 !2.59 3.66 !5.54 9.75 !24.35 18.00

(

))))))))))*

Fig. 3.4. Examples of di!erentiation matrices, Dr, for orders N = 1, 2, 4 in the toprow and N = 8 in the bottom row.

Examples of di!erentiation matrices for di!erent orders of approximationare shown in Fig. 3.4. A few observations are worth making regarding theseoperators. Inspection of Fig. 3.4 shows that

Dr,(i,j) = !Dr,(N!i,N!j),

known as skew-antisymmetric. This is a general property as long as ri = rN!i

(i.e., the interpolation points are symmetric around r = 0) and this can be ex-plored to compute fast matrix-vector products [291]. Furthermore, we noticethat each row-sum is exactly zero, reflecting that the derivative of a constantis zero. This also implies that Dr has at least one zero eigenvalue. A morecareful analysis will in fact show that Dr is nilpotent; that is, it has N + 1zero eigenvalues and only one eigenvector. Thus, Dr is not diagonizable butis similar only to a rank N + 1 Jordan block. These and other details arediscussed in depth in [159].

Example 3.2. To illustrate the accuracy of derivatives computed using Dr, letus consider a few examples.

We first consider the following analytic function

u(x) = exp(sin(!x)), x " [!1, 1],

and compute the errors of the computed derivatives as a function of the orderof the approximation, N . In Fig. 3.5 we show the L2-error as a function ofN and observe an exponentially fast decay of the error, known as spectralconvergence [159]. This is a clear indication of the strength of very high-ordermethods for the approximation of smooth functions.


0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2

Fig. 3.5. On the left, we show the global L2-error for the derivative of the smoothfunction in Example 3.2, highlighting exponential convergence. On the right, we showthe global L2-error for the derivative of the functions in Example 3.2, illustratingalgebraic convergence closely connected to the regularity of the function.

To o!er a more complete picture of the accuracy of the discrete derivative,let us also consider the sequence of functions

u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .

Note that u(i+1) # Ci; that is, u(1) is continuous but its derivative is discon-tinuous. In Fig. 3.5 we show the L2-error for the discrete derivative of u(i) fori = 1, 2, 3. We now observe a more moderate convergence rate as

""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.

If we recall that #u(i)

x

$

n= u(i!1)

n $ 1ni

then a rough estimate follows directly since""""

du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,

confirming the close connection between accuracy and smoothness indicatedin Fig. 3.5. We shall return to a more detailed discussion of these aspects inChapter 4 where we shall also realize that things are a bit more complex thanthe above result indicates.

To complete the discussion of the local operators, we consider the remain-ing local operator that is responsible for extracting the surface terms of theform


0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,



Spectral convergence


0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,



Accuracy ~ regularity

The local approximation - example

Consider


!!0.50 0.50!0.50 0.50

",

#!1.50 2.00 !0.50!0.50 0.00 0.50

0.50 !2.00 1.50

$,

%

&&&'

!5.00 6.76 !2.67 1.41 !0.50!1.24 0.00 1.75 !0.76 0.26

0.38 !1.34 0.00 1.34 !0.38!0.26 0.76 !1.75 0.00 1.24

0.50 !1.41 2.67 !6.76 5.00

(

)))*,

%

&&&&&&&&&&'

!18.00 24.35 !9.75 5.54 !3.66 2.59 !1.87 1.28 !0.50!4.09 0.00 5.79 !2.70 1.67 !1.15 0.82 !0.56 0.22

0.99 !3.49 0.00 3.58 !1.72 1.08 !0.74 0.49 !0.19!0.44 1.29 !2.83 0.00 2.85 !1.38 0.86 !0.55 0.21

0.27 !0.74 1.27 !2.66 0.00 2.66 !1.27 0.74 !0.27!0.21 0.55 !0.86 1.38 !2.85 0.00 2.83 !1.29 0.44

0.19 !0.49 0.74 !1.08 1.72 !3.58 0.00 3.49 !0.99!0.22 0.56 !0.82 1.15 !1.67 2.70 !5.79 0.00 4.09

0.50 !1.28 1.87 !2.59 3.66 !5.54 9.75 !24.35 18.00

(

))))))))))*

Fig. 3.4. Examples of di!erentiation matrices, Dr, for orders N = 1, 2, 4 in the toprow and N = 8 in the bottom row.

Examples of di!erentiation matrices for di!erent orders of approximationare shown in Fig. 3.4. A few observations are worth making regarding theseoperators. Inspection of Fig. 3.4 shows that

Dr,(i,j) = !Dr,(N!i,N!j),

known as skew-antisymmetric. This is a general property as long as ri = rN!i

(i.e., the interpolation points are symmetric around r = 0) and this can be ex-plored to compute fast matrix-vector products [291]. Furthermore, we noticethat each row-sum is exactly zero, reflecting that the derivative of a constantis zero. This also implies that Dr has at least one zero eigenvalue. A morecareful analysis will in fact show that Dr is nilpotent; that is, it has N + 1zero eigenvalues and only one eigenvector. Thus, Dr is not diagonizable butis similar only to a rank N + 1 Jordan block. These and other details arediscussed in depth in [159].

Example 3.2. To illustrate the accuracy of derivatives computed using Dr, letus consider a few examples.

We first consider the following analytic function

u(x) = exp(sin(!x)), x " [!1, 1],

and compute the errors of the computed derivatives as a function of the orderof the approximation, N . In Fig. 3.5 we show the L2-error as a function ofN and observe an exponentially fast decay of the error, known as spectralconvergence [159]. This is a clear indication of the strength of very high-ordermethods for the approximation of smooth functions.


0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,




0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,



Spectral convergence


0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,




0 16 32 48 6410!14

10!12

10!10

10!8

10!6

10!4

10!2

100

N

||ux !

D u h

||

101 102 10310!7

10!6

10!5

10!4

10!3

10!2

10!1

100

N

||ux !

D u h

||

u(1)

u(2)

u(3)

N!1/2

N!3/2

N!5/2



u(0)(x) =!! cos(!x), !1 " x < 0

cos(!x), 0 " x " 1,du(i+1)

dx= u(i), i = 0, 1, 2, 3 . . . .


""""du(i)

dx!Dru

(i)h

""""!

$ N1/2!i.


x

$

n= u(i!1)

n $ 1ni


du(i)

dx!Dru

(i)h

""""2

!

""%

n=N+1

1n2i

" 1N2i!1

,



Accuracy ~ regularity


Getting the grid and metric together

From a grid generator we will get

‣ A numbered set of vertices - VX‣ A set of elements, each of two vertices - EToV(K,2)

From this we need

‣ Grid ‣ Element sizes‣ Element normals and Jacobians‣ Element connectivity element-to-element edge-to-edge



We first compute the grid

3.3 Getting the grid together and computing the metric 57

If we specify the order, N , of the local approximations, we can computethe corresponding LGL nodes, r, in the reference element, I, and map themto each of the physical elements using the a!ne mapping, Eq. (3.1), as

>> va = EToV(:,1)’; vb = EToV(:,2)’;>> x = ones(Np,1)*VX(va) + 0.5*(r+1)*(VX(vb)-VX(va));

The array, x, is Np ! K and contains the physical coordinates of the gridpoints. The metric of the mapping is

rx =1xr

=1J

=1

Drx,

as implemented in GeometricFactors1D.m. Here, J represents the local trans-formation Jacobian.

GeometricFactors1D.m

function [rx,J] = GeometricFactors1D(x,Dr)

% function [rx,J] = GeometricFactors1D(x,Dr)% Purpose : Compute the metric elements for the local mappings% of the 1D elements

xr = Dr*x; J = xr; rx = 1./J;return

To readily access the elements along the faces (i.e., the two end points ofeach element), we form a small array with these two indices:

>> fmask1 = find( abs(r+1) < NODETOL)’;>> fmask2 = find( abs(r-1) < NODETOL)’;>> Fmask = [fmask1;fmask2]’;>> Fx = x(Fmask(:), :);

where the latter, Fx, is a 2 ! K array containing the physical coordinates ofthe edge nodes.

We now build the final piece of local information: the local outwardpointing normals and the inverse Jacobian at the interfaces, as done inNormals1D.m.

Normals1D.m

function [nx] = Normals1D

% function [nx] = Normals1D% Purpose : Compute outward pointing normals at elements faces

Globals1D;nx = zeros(Nfp*Nfaces, K);

and the local metric

The local normals are simple: nl = −1, nr = 1


We first compute the grid





rx =1xr

=1J

=1

Drx,










Normals1D.m








rx =1xr

=1J

=1

Drx,










Normals1D.m




and the local metric

The local normals are simple: nl = −1, nr = 1

Putting the pieces together in a code

Mesh tables:

EToV = [1 2; 2 3];VX = [0 0.5 1.0];

Global node numbering defined:

vmapM = [1 4 5 8];vmapP = [1 5 4 8];

Face node numbering defined:

mapM = [1 2 3 4];mapP = [1 3 2 4];mapI = [1];mapO = [4];

Outward point face normals:

nx = [-1 1; -1 1];

44 / 58


Mesh tables:

EToV = [1 2; 2 3];VX = [0 0.5 1.0];


vmapM = [1 4 5 8];vmapP = [1 5 4 8];




nx = [-1 1; -1 1];

44 / 58



We shall understand how to get the connectivitythrough an example - K=4, 5 vertices


% Define outward normalsnx(1, :) = -1.0; nx(2, :) = 1.0;return

>> Fscale = 1./(J(Fmask,:));

So far, all the elements are disconnected and we need to obtain informationabout which vertex on each element connects to which vertex on neighboringelements to form the global grid. If a vertex is not connected, it will be assumedto be a boundary with the possible need to impose a boundary condition.

We form an integer matrix, FToV, which is (2K) ! Nv. The 2K reflectsthe total number of faces, each of which have been assigned a global number.This supplies a map between this global face numbering, running from 1 to2K, and the global vertex numbering, running from 1 to Nv.

To recover information about which global face connects to which globalface, we compute the companion matrix

FToF = (FToV)(FToV)T ,

which is a (2K) ! (2K) integer matrix with ones on the diagonal, reflectingthat each face connects to itself. Thus, a pure connection matrix is found byremoving the self-reference

FToF = (FToV)(FToV)T " I.

In each row, corresponding to a particular face, one now finds either a value of1 in one column corresponding to the connecting face or only zeros, reflectingthat the particular face is an unconnected face (i.e., a boundary point).

Let us consider a simple example to make this approach clearer.

Example 3.3. Assume that we have read in the two arrays

V X =!0.0, 0.5, 1.5, 3.0, 2.5

", EToV =

#

$$%

1 22 33 55 4

&

''( ,

representing a grid with K = 4 elements and Nv = 5 vertices. We form

FToV =

#

$$$$$$$$$$%

1 0 0 0 00 1 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 0 10 0 0 1 00 0 0 0 1

&

''''''''''(

.

Form the companion matrix - FaceToVertex


% Define outward normalsnx(1, :) = -1.0; nx(2, :) = 1.0;return

>> Fscale = 1./(J(Fmask,:));

So far, all the elements are disconnected and we need to obtain informationabout which vertex on each element connects to which vertex on neighboringelements to form the global grid. If a vertex is not connected, it will be assumedto be a boundary with the possible need to impose a boundary condition.

We form an integer matrix, FToV, which is (2K) ! Nv. The 2K reflectsthe total number of faces, each of which have been assigned a global number.This supplies a map between this global face numbering, running from 1 to2K, and the global vertex numbering, running from 1 to Nv.

To recover information about which global face connects to which globalface, we compute the companion matrix

FToF = (FToV)(FToV)T ,

which is a (2K) ! (2K) integer matrix with ones on the diagonal, reflectingthat each face connects to itself. Thus, a pure connection matrix is found byremoving the self-reference

FToF = (FToV)(FToV)T " I.

In each row, corresponding to a particular face, one now finds either a value of1 in one column corresponding to the connecting face or only zeros, reflectingthat the particular face is an unconnected face (i.e., a boundary point).

Let us consider a simple example to make this approach clearer.

Example 3.3. Assume that we have read in the two arrays

V X =!0.0, 0.5, 1.5, 3.0, 2.5

", EToV =

#

$$%

1 22 33 55 4

&

''( ,

representing a grid with K = 4 elements and Nv = 5 vertices. We form

FToV =

#

$$$$$$$$$$%

1 0 0 0 00 1 0 0 00 1 0 0 00 0 1 0 00 0 1 0 00 0 0 0 10 0 0 1 00 0 0 0 1

&

''''''''''(

.

Faces

Element 1

Element 2

Element 3Element 4


Getting the grid and metric together3.3 Getting the grid together and computing the metric 59

to connect the 2K local faces to the Nv globally numbered faces.The connectivity can then be recovered from the operator

(FToV)(FToV)T =

!

""""""""""#

1 0 0 0 0 0 0 00 1 1 0 0 0 0 00 1 1 0 0 0 0 00 0 0 1 1 0 0 00 0 0 1 1 0 0 00 0 0 0 0 1 0 10 0 0 0 0 0 1 00 0 0 0 0 1 0 1

$

%%%%%%%%%%&

.

The identity diagonal reflects the self-reference. Furthermore, we see thatbeyond the diagonal, rows 1 and 7 indicate that these vertices have no con-nection, and are assumed to reflect the outer boundaries of the computationaldomain. Finally, we see that face 2 connects to face 3 and face 3 to face 2, inagreement with the basic list of input data.

From the FToF-matrix we can immediately recover the element-to-element,EToE, and element-to-face, EToF, information as each segment of two rows orcolumns corresponds to one element and its two faces. The whole procedureis implemented in Connect1D.m.

Connect1D.m

function [EToE, EToF] = Connect1D(EToV)

% function [EToE, EToF] = Connect1D(EToV)% Purpose : Build global connectivity arrays for 1D grid based% on standard EToV input array from grid generator

Nfaces = 2;% Find number of elements and verticesK = size(EToV,1); TotalFaces = Nfaces*K; Nv = K+1;

% List of local face to local vertex connectionsvn = [1,2];

% Build global face to node sparse arraySpFToV = spalloc(TotalFaces, Nv, 2*TotalFaces);sk = 1;for k=1:Kfor face=1:Nfaces

SpFToV( sk, EToV(k, vn(face))) = 1;sk = sk+1;

endend

Consider

From this all connectivity information can be found

Edge 1 only self connects -- it is a boundary ! Edge 2 connects to Edge 3 Edge 3 connects to Edge 2 etc

Element 1

Element 2

Element 3

Element 4



This is used to built index maps

‣ vmapM -- such that u(vmapM) = interior u‣ vmapP -- such that u(vmapP) = exterior u‣ vmapB -- such that u(vmapB) = boundary u


Mesh tables:

EToV = [1 2; 2 3];VX = [0 0.5 1.0];


vmapM = [1 4 5 8];vmapP = [1 5 4 8];




nx = [-1 1; -1 1];

44 / 58


The global scheme

To connect all elements and compute the metric we need routines like

Element-wise operations

A summary of scripts needed for preprocessing for 1D DG-FEMcomputations in Matlab

Globals1D Define list of globals variablesMeshGen1D Generates a simple equidistant grid with K elementsStartup1D Main script for pre-processingBuildMaps1D Automatically create index maps from conn. and bc tablesNormals1D Compute outward pointing normals at elements facesConnect1D Build global connectivity arrays for 1D gridGeometricFactors1D Compute metrics of local mappingsLift1D Compute surface integral term in DG formulation

! Components listed here are general and reusable for each solver - enables fastproto-typing.

! MeshGen1D can be susbstituted with your own preferred meshinterface/generator.

40 / 58

These are also problem independent !


Putting it together - an example

Consider again the wave equation


Table 3.2. Coe!cients for the low-storage five-stage fourth-order ERK method(LSERK).

i ai bi ci

1 014329971744779575080441755

0

2 ! 5673018057731357537059087

516183667771713612068292357

14329971744779575080441755

3 !24042679903932016746695238

17201463215492090206949498

25262693414296820363962896

4 !35509186866462091501179385

31345643535374481467310338

20063455193173224310063776

5 !1275806237668842570457699

227782119143714882151754819

28023216131382924317926251

alternative to this is a low-storage version [46] of the fourth-order method(LSERK) of the form

p(0) = un,

i ! [1, . . . , 5] :

!k(i) = aik

(i!1) + !tLh

"p(i!1), tn + ci!t

#,

p(i) = p(i!1) + bik(i),

un+1h = p(5). (3.5)

The coe!cients needed in the LSERK are given in Table 3.2. The main di"er-ence here is that only one additional storage level is required, thus reducingthe memory usage significantly. On the other hand, this comes at the price ofan additional function evaluation, as the low-storage version has five stages.

At first, it would seem that the additional stage makes the low-storageapproach less interesting due to the added cost. However, as we will discussin more detail in Chapter 4, the added cost in the low-storage RK is o"set byallowing a larger stable timestep, !t.

It should be emphasized that these methods are not exclusive and manyalternatives exist [40, 143, 144]. In particular, for strongly nonlinear problems,the added nonlinear stability of strong stability-preserving methods may beadvantageous, as we will discuss further in Chapter 5.

3.5 Putting it all together

Let us now return to the simple linear wave equation

"u

"t+ 2#

"u

"x= 0, x ! [0, 2#],

u(x, 0) = sin(x),u(0, t) = " sin(2#t).

Following the recipe we have developed, we get

3.5 Putting it all together 65

It is easy to see that the exact solution is

u(x, t) = sin(x ! 2!t).

Following the developments in Chapter 2, we assume the local solution is wellapproximated as

ukh(x, t) =

Np!

i=1

ukh(xk

i , t)"ki (x),

and the combination of these local solutions as the approximation to the globalsolution. This yields the local semidiscrete scheme

Mk dukh

dt+ 2!Suk

h ="!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,

orduk

h

dt+ 2!(Mk)"1Suk

h = (Mk)"1"!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,

The analysis in Example 2.3 shows that choosing the flux as

(2!u)! = 2!u + 2!1 ! #

2[[u]], 0 " # " 1

results in a stable scheme.An implementation of this is shown in AdvecRHS1D.m. The parameter #

can be used to adjust the flux; for example # = 1 is a central flux and # = 0reflects an upwind flux.

AdvecRHS1D.m

function [rhsu] = AdvecRHS1D(u,time, a)

% function [rhsu] = AdvecRHS1D(u,time)% Purpose : Evaluate RHS flux in 1D advection

Globals1D;

% form field differences at facesalpha=1;du = zeros(Nfp*Nfaces,K);du(:) = (u(vmapM)-u(vmapP)).*(a*nx(:)-(1-alpha)*abs(a*nx(:)))/2;

% impose boundary condition at x=0uin = -sin(a*time);du (mapI) = (u(vmapI)- uin ).*(a*nx(mapI)-(1-alpha)*abs(a*nx(mapI)))/2;du (mapO) = 0;

% compute right hand sides of the semi-discrete PDErhsu = -a*rx.*(Dr*u) + LIFT*(Fscale.*(du));return



u(x, t) = sin(x ! 2!t).


ukh(x, t) =

Np!

i=1

ukh(xk

i , t)"ki (x),


Mk dukh

dt+ 2!Suk

h ="!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,

orduk

h

dt+ 2!(Mk)"1Suk

h = (Mk)"1"!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,


(2!u)! = 2!u + 2!1 ! #

2[[u]], 0 " # " 1



AdvecRHS1D.m



Globals1D;




and the stable flux


Putting it together

We always formulate it through 3 partsPutting the pieces together in a codeTo build your own solver using the DGFEM codes

AdvecDriver1D Matlab main function for solving the 1D advection equation.Advec1D(u,FinalTime) Matlab function for time integration of the semidiscrete PDE.AdvecRHS1D(u,time,a) Matlab function defining right hand side of semidiscrete PDE

DGFEM Toolbox\1D

AdvecDriver1D Advec1D AdvecRHS1D

! Several examples, functions, utilities in DGFEM package! Fast proto-typing! Write your own solver independent on mesh

39 / 58





u(x, t) = sin(x ! 2!t).


ukh(x, t) =

Np!

i=1

ukh(xk

i , t)"ki (x),


Mk dukh

dt+ 2!Suk

h ="!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,

orduk

h

dt+ 2!(Mk)"1Suk

h = (Mk)"1"!k(x)(2!uk

h ! (2!u)!)#xk

r

xkl

,


(2!u)! = 2!u + 2!1 ! #

2[[u]], 0 " # " 1



AdvecRHS1D.m



Globals1D;





Temporal integrationAs we have already discussed we use RK methods

3.4 Dealing with time 63

rk4a = [ 0.0 ...-567301805773.0/1357537059087.0 ...-2404267990393.0/2016746695238.0 ...-3550918686646.0/2091501179385.0 ...-1275806237668.0/842570457699.0];

rk4b = [ 1432997174477.0/9575080441755.0 ...5161836677717.0/13612068292357.0 ...1720146321549.0/2090206949498.0 ...3134564353537.0/4481467310338.0 ...2277821191437.0/14882151754819.0];

rk4c = [ 0.0 ...1432997174477.0/9575080441755.0 ...2526269341429.0/6820363962896.0 ...2006345519317.0/3224310063776.0 ...2802321613138.0/2924317926251.0];

3.4 Dealing with time

The emphasis so far has been on the spatial dimension and a discrete rep-resentation of this. This reflects a method-of-lines approach where we dis-cretize space and time separately, and use some standard technique to solvethe ordinary di!erential equations for the latter.

We follow this approach here and, furthermore, focus on the use of explicitRunge-Kutta (RK) methods for integration in the temporal dimension. Todiscretize the semidiscrete problem

duh

dt= Lh (uh, t) ,

where uh is the vector of unknowns, we can use the standard fourth-orderfour stage explicit RK method (ERK)

k(1) = Lh (unh, tn) ,

k(2) = Lh

!un

h +12!tk(1), tn +

12!t

",

k(3) = Lh

!un

h +12!tk(2), tn +

12!t

",

k(4) = Lh

#un

h + !tk(3), tn + !t$

,

un+1h = un

h +16!t

#k(1) + 2k(2) + 2k(3) + k(4)

$, (3.4)

to advance from unh to un+1

h , separated by the timestep, !t.As simple and widely used as this classic ERK approach is, it has the

disadvantage that it requires four extra storage arrays, k(i). An attractive



i ai bi ci

1 014329971744779575080441755

0

2 ! 5673018057731357537059087

516183667771713612068292357

14329971744779575080441755

3 !24042679903932016746695238

17201463215492090206949498

25262693414296820363962896

4 !35509186866462091501179385

31345643535374481467310338

20063455193173224310063776

5 !1275806237668842570457699

227782119143714882151754819

28023216131382924317926251


p(0) = un,

i ! [1, . . . , 5] :

!k(i) = aik

(i!1) + !tLh

"p(i!1), tn + ci!t

#,

p(i) = p(i!1) + bik(i),

un+1h = p(5). (3.5)






"u

"t+ 2#

"u

"x= 0, x ! [0, 2#],

u(x, 0) = sin(x),u(0, t) = " sin(2#t).



i ai bi ci

1 014329971744779575080441755

0

2 ! 5673018057731357537059087

516183667771713612068292357

14329971744779575080441755

3 !24042679903932016746695238

17201463215492090206949498

25262693414296820363962896

4 !35509186866462091501179385

31345643535374481467310338

20063455193173224310063776

5 !1275806237668842570457699

227782119143714882151754819

28023216131382924317926251


p(0) = un,

i ! [1, . . . , 5] :

!k(i) = aik

(i!1) + !tLh

"p(i!1), tn + ci!t

#,

p(i) = p(i!1) + bik(i),

un+1h = p(5). (3.5)






"u

"t+ 2#

"u

"x= 0, x ! [0, 2#],

u(x, 0) = sin(x),u(0, t) = " sin(2#t).

There are many suitable alternatives


Putting it together - an example66 3 Making it work in one dimension

To complete the scheme for solving the advection equation, we need onlyintegrate the system in time. As discussed in Section 3.4, we do this using alow-storage RK method. This is implemented in Advec1D.m.

Advec1D.m

function [u] = Advec1D(u, FinalTime)

% function [u] = Advec1D(u, FinalTime)% Purpose : Integrate 1D advection until FinalTime starting with% initial the condition, u

Globals1D;time = 0;

% Runge-Kutta residual storageresu = zeros(Np,K);

% compute time step sizexmin = min(abs(x(1,:)-x(2,:)));CFL=0.75; dt = CFL/(2*pi)*xmin; dt = .5*dt;Nsteps = ceil(FinalTime/dt); dt = FinalTime/Nsteps;

% advection speeda = 2*pi;

% outer time step loopfor tstep=1:Nsteps

for INTRK = 1:5timelocal = time + rk4c(INTRK)*dt;[rhsu] = AdvecRHS1D(u, timelocal, a);resu = rk4a(INTRK)*resu + dt*rhsu;u = u+rk4b(INTRK)*resu;

end;% Increment timetime = time+dt;

end;return

One issue we have not discussed yet is how to choose the timestep to ensurea discretely stable scheme. In Advec1D.m we use the guideline

!t ! C

2"mink,i

!xki ,

where !xki = xk

i+1"xki and 2" is the maximum wave speed. The constant, C,

is a Courant-Friedrichs-Levy (CFL)-like constant that can be expected to beO(1). We will discuss this in more detail in Chapter 4 but the above expressionis an excellent guideline.



3.6 Maxwell’s equations 67

Finally, we need a driver routine where the grid information is read in, theglobal grid and the metric are assembled as discussed in Section 3.3, and theinitial conditions are set. An example of this is shown in AdvecDriver1D.m.All results presented in Chapter 2 have been obtained using this simple code.

AdvecDriver1D.m

% Driver script for solving the 1D advection equationsGlobals1D;

% Order of polymomials used for approximationN = 8;

% Generate simple mesh[Nv, VX, K, EToV] = MeshGen1D(0.0,2.0,10);

% Initialize solver and construct grid and metricStartUp1D;

% Set initial conditionsu = sin(x);

% Solve ProblemFinalTime = 10;[u] = Advec1D(u,FinalTime);

It is worth emphasizing that the above three routines, AdvecRHS1D.m,Advec1D.m, and AdvecDriver1D.m, are all that is needed to solve the problemat any order and using any grid. In fact, only AdvecRHS1D.m requires sub-stantial changes to solve another problem as we will see next, where we discussthe routines for a more complex problem. The grid information is generatedin MeshGen1D.m, which is described in Appendix B.

3.6 Maxwell’s equations

As a slightly more complicated problem, let us consider the one-dimensionalMaxwell’s equations

!(x)"E

"t= !"H

"x, µ(x)

"H

"t= !"E

"x, (3.6)

where (E,H) represent the electric and magnetic fields, respectively, and thematerial parameters, !(x) and µ(x), reflect the electric permittivity and mag-netic permeability, respectively.

We solve these equations in a fixed domain x " [!2, 2] with both !(x) andµ(x) changing discontinuously at x = 0. For simplicity, we assume that the

Structure is the same for any time-dependent problem.

Often only the first routine needs any work.



2.2 Basic elements of the schemes 29

0 0.25 0.5 0.75 1!1.25

!1

!0.75

!0.5

!0.25

0

0.25

0.5

0.75

1

1.25

x/2!

u(x,2

!)

0 0.25 0.5 0.75 1!1.25

!1

!0.75

!0.5

!0.25

0

0.25

0.5

0.75

1

1.25

x/2!

u(x,2!)

Fig. 2.2. On the left we compare the solution to the wave equation, computedusing an N = 1, K = 12 discontinuous Galerkin method with a central flux. On theright the same computation employs an upwind flux. In both cases, the dashed linerepresents the exact solution.

and K = 12 elements, we show the solution at T = 2!, computed using acentral flux. We note in particular the discontinuous nature of the solution,which is a characteristic of the family of methods discussed here. To contrastthis, we show on the right of Fig. 2.2 the same solution computed using apure upwind flux. This leads to a solution with smaller jumps between theelements. However, the dissipative nature of the upwind flux is also apparent.An important lesson to learn from this is that a visually smoother solutionis not necessarily a more accurate solution, although in the case consideredhere, the global errors are comparable.

Many of the observations made in the above example regarding high-ordermethods can be put on firmer ground through an analysis of the phase errors.Although we will return to some of this again in Chapter 4, other insightfuldetails can be found in [155, 159, 208, 307].

The example illustrates that to solve a given problem to a specific accuracy,one is most likely best o! by having the ability to choose the element size, h,as well as the order of the scheme, N , independently, and preferably in a localmanner. The ability to do this is one of the main advantages of the family ofschemes discussed in this text.

2.2.2 An alternative viewpoint

Before we continue, it is instructive to consider an alternative derivation. Themain dilemma posed by the choice of a piecewise polynomial representationof uh with no a priori assumptions about continuity is how to evaluate agradient. Starting with a solution defined on each of the elements of the grid,uk

h, we can imagine continuing the function from the boundaries and beyondthe element. In the one-dimensional case, this is achieved by adding two scaledHeaviside functions, defined as


22 2 The key ideas

Example 2.4. Consider Eq. (2.1) as

∂u

∂t− 2π

∂u

∂x= 0, x ∈ [0, 2π],

with periodic boundary conditions and initial condition as

u(x, 0) = sin(lx), l =2π

λ,

where λ is the wavelength. We use the strong form, Eq. (2.8), although for this simple example, theweak form yields identical results. The nodes are chosen as the Legendre-Gauss-Lobatto nodes as weshall discuss in detail in Chapter 3. An upwind flux is used and a fourth-order explicit Runge-Kuttamethod is employed to integrate the equations in time with the timestep chosen small enough toensure that timestep errors can be neglected (See Chapter 3 for details on the implementation).

In Table 2.1 we list a number of results, showing the global L2-error at final time T = π as afunction of the number of elements, K, and the order of the local approximation, N . Inspectingthese results, we observe several things. First, the scheme is clearly convergent and there are tworoads to a converged result; one can increase the local order of approximation, N , and/or one canincrease the number of elements, K.

Table 2.1. Global L2-errors when solving the wave equation using K elements each with a local orderof approximation, N . Note that for N = 8, the finite precision dominates and destroys the expectedconvergence rate.

N\ K 2 4 8 16 32 64 Convergence rate1 – 4.0E-01 9.1E-02 2.3E-02 5.7E-03 1.4E-03 2.02 2.0E-01 4.3E-02 6.3E-03 8.0E-04 1.0E-04 1.3E-05 3.04 3.3E-03 3.1E-04 9.9E-06 3.2E-07 1.0E-08 3.3E-10 5.08 2.1E-07 2.5E-09 4.8E-12 2.2E-13 5.0E-13 6.6E-13 9.0

The rate by which the results converge are not, however, the same when changing N and K. Ifwe define h = 2π/K as a measure of the size of the local element, we observe that

u− uhΩ,h ≤ ChN+1.

Thus, it is the order of the local approximation that gives the fast convergence rate. The constant,C, does not depend on h, but it may depend on the final time, T , of the solution. To highlight this,we consider in Table 2.2 the same problem but solved at different final times, T .

This indicates a linear scaling in time as

u− uhΩ,h ≤ C(T )hN+1 (c1 + c2T )hN+1,

Clearly, c1 and c2 are problem-dependent constants.The high accuracy reflected in Table 2.1, however, comes at a price. In Table 2.3 we show the

approximate execution times, scaled to one for (N,K) = (1, 2), for all the examples in Table 2.1.The execution time and, thus, the computational effort scales approximately like

2.2 Basic elements of the schemes 29

0 0.25 0.5 0.75 1!1.25

!1

!0.75

!0.5

!0.25

0

0.25

0.5

0.75

1

1.25

x/2!

u(x,2

!)

0 0.25 0.5 0.75 1!1.25

!1

!0.75

!0.5

!0.25

0

0.25

0.5

0.75

1

1.25

x/2!

u(x,2!)

Fig. 2.2. On the left we compare the solution to the wave equation, computedusing an N = 1, K = 12 discontinuous Galerkin method with a central flux. On theright the same computation employs an upwind flux. In both cases, the dashed linerepresents the exact solution.

and K = 12 elements, we show the solution at T = 2!, computed using acentral flux. We note in particular the discontinuous nature of the solution,which is a characteristic of the family of methods discussed here. To contrastthis, we show on the right of Fig. 2.2 the same solution computed using apure upwind flux. This leads to a solution with smaller jumps between theelements. However, the dissipative nature of the upwind flux is also apparent.An important lesson to learn from this is that a visually smoother solutionis not necessarily a more accurate solution, although in the case consideredhere, the global errors are comparable.

Many of the observations made in the above example regarding high-ordermethods can be put on firmer ground through an analysis of the phase errors.Although we will return to some of this again in Chapter 4, other insightfuldetails can be found in [155, 159, 208, 307].

The example illustrates that to solve a given problem to a specific accuracy,one is most likely best o! by having the ability to choose the element size, h,as well as the order of the scheme, N , independently, and preferably in a localmanner. The ability to do this is one of the main advantages of the family ofschemes discussed in this text.

2.2.2 An alternative viewpoint

Before we continue, it is instructive to consider an alternative derivation. Themain dilemma posed by the choice of a piecewise polynomial representationof uh with no a priori assumptions about continuity is how to evaluate agradient. Starting with a solution defined on each of the elements of the grid,uk

h, we can imagine continuing the function from the boundaries and beyondthe element. In the one-dimensional case, this is achieved by adding two scaledHeaviside functions, defined as


A slightly more complicated example

Consider linear system

uv

t

+

0 ab 0

uv

x

= 0

How do we choose the flux in this case ?

Clearly ab>0 for hyperbolicity

Recall that A =

0 ab 0

S =

1√b− 1√

b1√a

1√a

S−1 = 1

2

√b

√a

−√

b√

a

Λ =

√ab 00 −

√ab

S−1AS = Λ

With



This allows us to transform the equation as

S−1

uv

t

+ ΛS−1

uv

x

= 0

So now we know that 12 (√

bu +√

av)

12 (−

√bu +

√av)

propagates right

propagates left

SΛ+

√bu−+

√av−

2 + Λ−−√

bu++√

av+

2

Λ = Λ+ + Λ−

This suggests a flux like



√a

2

√a(v+ + v−) +

√b(u− − u+)

√b

2

√b(u+ + u−) +

√a(v− − v+)

av +√

ab2 n · [[u]]

bu +√

ab2 n · [[v]]

uv

t

+

0 ab 0

uv

x

= 0

Straightforward manipulations yield

or the equivalent

Recall

we recognize this as the LF flux


Shallow water equations

Consider 1D shallow water equations in periodic domain

Examples: error behaviorConsider the linear shallow water equations in one horizontaldimension on a periodic domain

!

!t

!

"u

"

=

!

0 !h

!g 0

"

!

!x

!

"u

"

Tests of h! and p-refinement

101 102 10310−12

10−10

10−8

10−6

10−4

10−2

Number of degrees of freedom

Erro

r

13

1

5

1

7h−version (P=2)h−version (P=4)h−version (P=6)

0 50 100 150 200 250 300 350

10−10

10−5

100


Erro

r

h−version (P=1)p−version (K=20)

Again, the error behaves as

||u ! uh||!,h " ChN+1

35 / 41

Tests of h- and p-refinement

Examples: error behaviorConsider the linear shallow water equations in one horizontaldimension on a periodic domain

!

!t

!

"u

"

=

!

0 !h

!g 0

"

!

!x

!

"u

"

Tests of h! and p-refinement

101 102 10310−12

10−10

10−8

10−6

10−4

10−2


Erro

r

13

1

5

1

7h−version (P=2)h−version (P=4)h−version (P=6)

0 50 100 150 200 250 300 350

10−10

10−5

100


Erro

r

h−version (P=1)p−version (K=20)

Again, the error behaves as

||u ! uh||!,h " ChN+1

35 / 41

Examples: error behavior

Consider the simple advection equation on a periodic domain

!tu ! 2"!xu = 0, x " [0, 2"], u(x , 0) = sin(lx), l = 2!"

Exact solution is then u(x , t) = sin(l(x ! 2"t))).

Errors at final time T = ".

N\ K 2 4 8 16 32 64 Convergence rate1 - 4.0E-01 9.1E-02 2.3E-02 5.7E-03 1.4E-03 2.02 2.0E-01 4.3E-02 6.3E-03 8.0E-04 1.0E-04 1.3E-05 3.04 3.3E-03 3.1E-04 9.9E-06 3.2E-07 1.0E-08 3.3E-10 5.08 2.1E-07 2.5E-09 4.8E-12 2.2E-13 5.0E-13 6.6E-13 != 9.0

Error is seen to behave as

||u ! uh||!,h # ChN+1

Clearly, paths to convergence are based on adjusting the size ofelements (h-convergence), the polynomial order (p-convergence) orcombinations hereo!.

34 / 41


Putting it to use - an exampleConsider Maxwell’s equations


Finally, we need a driver routine where the grid information is read in, theglobal grid and the metric are assembled as discussed in Section 3.3, and theinitial conditions are set. An example of this is shown in AdvecDriver1D.m.All results presented in Chapter 2 have been obtained using this simple code.

AdvecDriver1D.m

% Driver script for solving the 1D advection equationsGlobals1D;

% Order of polymomials used for approximationN = 8;

% Generate simple mesh[Nv, VX, K, EToV] = MeshGen1D(0.0,2.0,10);


% Set initial conditionsu = sin(x);

% Solve ProblemFinalTime = 10;[u] = Advec1D(u,FinalTime);

It is worth emphasizing that the above three routines, AdvecRHS1D.m,Advec1D.m, and AdvecDriver1D.m, are all that is needed to solve the problemat any order and using any grid. In fact, only AdvecRHS1D.m requires sub-stantial changes to solve another problem as we will see next, where we discussthe routines for a more complex problem. The grid information is generatedin MeshGen1D.m, which is described in Appendix B.

3.6 Maxwell’s equations

As a slightly more complicated problem, let us consider the one-dimensionalMaxwell’s equations

!(x)"E

"t= !"H

"x, µ(x)

"H

"t= !"E

"x, (3.6)

where (E,H) represent the electric and magnetic fields, respectively, and thematerial parameters, !(x) and µ(x), reflect the electric permittivity and mag-netic permeability, respectively.

We solve these equations in a fixed domain x " [!2, 2] with both !(x) andµ(x) changing discontinuously at x = 0. For simplicity, we assume that the

-1 10

ε1, µ1 ε2, µ2

BC: E(-1,t)=E(1,t)=0 -- a PEC cavity

Interface conditions: Continuity of fields

The problem has a nice exact solution


Putting it to use - an example

Scheme becomes


materials are otherwise piecewise constant. Relaxing this would only requireminor modifications. Furthermore, we assume that E(!2, 0) = E(2, 0) = 0,corresponding to a physical situation where the computational domain isenclosed in a metallic cavity.

To develop the scheme, we take the usual path and seek an approximation(E,H) " (Eh,Hh) being composed as the direct sum of K local polynomials(Ek

h,Hkh) on the form

!Ek

h(x, t)Hk

h(x, t)

"=

Np#

i=1

!Ek

h(xki , t)

Hkh(xk

i , t)

"!ki (x).

Introducing this into Eq. (3.6) and requiring Maxwell’s equations to be satis-fied locally on the strong discontinuous Galerkin form yields the semidiscretescheme

dEkh

dt+

1Jk"k

DrHkh =

1Jk"k

M!1$!k(x)(Hk

h ! H")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Hkh ! H")!k(x) dx,

dHkh

dt+

1Jkµk

DrEkh =

1Jkµk

M!1$!k(x)(Ek

h ! E")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Ekh ! E")!k(x) dx.

As the fluxes, we will use those derived in Example 2.6 on the form

H" =1

Z

'ZH +

12[[E]]

(, E" =

1Y

'Y E +

12[[H]]

(,

where

Z± =

)µ±

"±= (Y ±)!1.

This yields the terms

H! ! H" =1

2Z*Z+[[H]] ! [[E]]

+,

E! ! E" =1

2Y *Y +[[E]] ! [[H]]

+,

as the penalty terms entering on the right-hand side of the semidiscretescheme. This is all implemented in MaxwellRHS1D.m.


Scheme becomes




h,Hkh) on the form

!Ek

h(x, t)Hk

h(x, t)

"=

Np#

i=1

!Ek

h(xki , t)

Hkh(xk

i , t)

"!ki (x).


dEkh

dt+

1Jk"k

DrHkh =

1Jk"k

M!1$!k(x)(Hk

h ! H")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Hkh ! H")!k(x) dx,

dHkh

dt+

1Jkµk

DrEkh =

1Jkµk

M!1$!k(x)(Ek

h ! E")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Ekh ! E")!k(x) dx.


H" =1

Z

'ZH +

12[[E]]

(, E" =

1Y

'Y E +

12[[H]]

(,

where

Z± =

)µ±

"±= (Y ±)!1.


H! ! H" =1

2Z*Z+[[H]] ! [[E]]

+,

E! ! E" =1

2Y *Y +[[E]] ! [[H]]

+,





h,Hkh) on the form

!Ek

h(x, t)Hk

h(x, t)

"=

Np#

i=1

!Ek

h(xki , t)

Hkh(xk

i , t)

"!ki (x).


dEkh

dt+

1Jk"k

DrHkh =

1Jk"k

M!1$!k(x)(Hk

h ! H")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Hkh ! H")!k(x) dx,

dHkh

dt+

1Jkµk

DrEkh =

1Jkµk

M!1$!k(x)(Ek

h ! E")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Ekh ! E")!k(x) dx.


H" =1

Z

'ZH +

12[[E]]

(, E" =

1Y

'Y E +

12[[H]]

(,

where

Z± =

)µ±

"±= (Y ±)!1.


H! ! H" =1

2Z*Z+[[H]] ! [[E]]

+,

E! ! E" =1

2Y *Y +[[E]] ! [[H]]

+,





h,Hkh) on the form

!Ek

h(x, t)Hk

h(x, t)

"=

Np#

i=1

!Ek

h(xki , t)

Hkh(xk

i , t)

"!ki (x).


dEkh

dt+

1Jk"k

DrHkh =

1Jk"k

M!1$!k(x)(Hk

h ! H")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Hkh ! H")!k(x) dx,

dHkh

dt+

1Jkµk

DrEkh =

1Jkµk

M!1$!k(x)(Ek

h ! E")%xk

r

xkl

=1

Jk"kM!1

& xkr

xkl

n · (Ekh ! E")!k(x) dx.


H" =1

Z

'ZH +

12[[E]]

(, E" =

1Y

'Y E +

12[[H]]

(,

where

Z± =

)µ±

"±= (Y ±)!1.


H! ! H" =1

2Z*Z+[[H]] ! [[E]]

+,

E! ! E" =1

2Y *Y +[[E]] ! [[H]]

+,



Putting it to use - an example3.6 Maxwell’s equations 69

MaxwellRHS1D.m

function [rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu)

% function [rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu)% Purpose : Evaluate RHS flux in 1D Maxwell

Globals1D;

% Compute impedanceZimp = sqrt(mu./eps);

% Define field differences at facesdE = zeros(Nfp*Nfaces,K); dE(:) = E(vmapM)-E(vmapP);dH = zeros(Nfp*Nfaces,K); dH(:) = H(vmapM)-H(vmapP);Zimpm = zeros(Nfp*Nfaces,K); Zimpm(:) = Zimp(vmapM);Zimpp = zeros(Nfp*Nfaces,K); Zimpp(:) = Zimp(vmapP);Yimpm = zeros(Nfp*Nfaces,K); Yimpm(:) = 1./Zimpm(:);Yimpp = zeros(Nfp*Nfaces,K); Yimpp(:) = 1./Zimpp(:);

% Homogeneous boundary conditions, Ez=0Ebc = -E(vmapB); dE (mapB) = E(vmapB) - Ebc;Hbc = H(vmapB); dH (mapB) = H(vmapB) - Hbc;

% evaluate upwind fluxesfluxE = 1./(Zimpm + Zimpp).*(nx.*Zimpp.*dH - dE);fluxH = 1./(Yimpm + Yimpp).*(nx.*Yimpp.*dE - dH);

% compute right hand sides of the PDE’srhsE = (-rx.*(Dr*H) + LIFT*(Fscale.*fluxE))./eps;rhsH = (-rx.*(Dr*E) + LIFT*(Fscale.*fluxH))./mu;return

The temporal integration is done using Maxwell1D.m which is essentiallyunchanged from Advec1D.m in the previous section. The only change is thatwe are now integrating a system and, thus, need to integrate all componentsof the system at each stage.

Maxwell1D.m

function [E,H] = Maxwell1D(E,H,eps,mu,FinalTime);

% function [E,H] = Maxwell1D(E,H,eps,mu,FinalTime)% Purpose : Integrate 1D Maxwell’s until FinalTime starting with% conditions (E(t=0),H(t=0)) and materials (eps,mu).

Globals1D;time = 0;

% Runge-Kutta residual storage


Putting it to use - an examplePutting it to use - an example


MaxwellRHS1D.m

function [rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu)

% function [rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu)% Purpose : Evaluate RHS flux in 1D Maxwell

Globals1D;

% Compute impedanceZimp = sqrt(mu./eps);

% Define field differences at facesdE = zeros(Nfp*Nfaces,K); dE(:) = E(vmapM)-E(vmapP);dH = zeros(Nfp*Nfaces,K); dH(:) = H(vmapM)-H(vmapP);Zimpm = zeros(Nfp*Nfaces,K); Zimpm(:) = Zimp(vmapM);Zimpp = zeros(Nfp*Nfaces,K); Zimpp(:) = Zimp(vmapP);Yimpm = zeros(Nfp*Nfaces,K); Yimpm(:) = 1./Zimpm(:);Yimpp = zeros(Nfp*Nfaces,K); Yimpp(:) = 1./Zimpp(:);

% Homogeneous boundary conditions, Ez=0Ebc = -E(vmapB); dE (mapB) = E(vmapB) - Ebc;Hbc = H(vmapB); dH (mapB) = H(vmapB) - Hbc;

% evaluate upwind fluxesfluxE = 1./(Zimpm + Zimpp).*(nx.*Zimpp.*dH - dE);fluxH = 1./(Yimpm + Yimpp).*(nx.*Yimpp.*dE - dH);

% compute right hand sides of the PDE’srhsE = (-rx.*(Dr*H) + LIFT*(Fscale.*fluxE))./eps;rhsH = (-rx.*(Dr*E) + LIFT*(Fscale.*fluxH))./mu;return

The temporal integration is done using Maxwell1D.m which is essentiallyunchanged from Advec1D.m in the previous section. The only change is thatwe are now integrating a system and, thus, need to integrate all componentsof the system at each stage.

Maxwell1D.m

function [E,H] = Maxwell1D(E,H,eps,mu,FinalTime);

% function [E,H] = Maxwell1D(E,H,eps,mu,FinalTime)% Purpose : Integrate 1D Maxwell’s until FinalTime starting with% conditions (E(t=0),H(t=0)) and materials (eps,mu).

Globals1D;time = 0;

% Runge-Kutta residual storage


resE = zeros(Np,K); resH = zeros(Np,K);

% compute time step sizexmin = min(abs(x(1,:)-x(2,:)));CFL=1.0; dt = CFL*xmin;Nsteps = ceil(FinalTime/dt); dt = FinalTime/Nsteps;


for INTRK = 1:5[rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu);

resE = rk4a(INTRK)*resE + dt*rhsE;resH = rk4a(INTRK)*resH + dt*rhsH;

E = E+rk4b(INTRK)*resE;H = H+rk4b(INTRK)*resH;

end% Increment timetime = time+dt;

endreturn

In the final routine, MaxwellDriver1D.m, the only significant di!erence fromAdvecDriver1D.m is the need to also specify the spatial distribution of ! andµ. In this particular case, we have assumed that these are constant in each el-ement but can jump between elements. Furthermore, we assume here that thejump is between the two middle elements. Clearly, this specification is problemdependent and can be changed as needed. Also, if ! and/or µ vary smoothlywithin the elements, this can be specified here with no further changes else-where.

MaxwellDriver1D.m

% Driver script for solving the 1D Maxwell’s equationsGlobals1D;

% Polynomial order used for approximationN = 6;

% Generate simple mesh[Nv, VX, K, EToV] = MeshGen1D(-2.0,2.0,80);


% Set up material parameterseps1 = [ones(1,K/2), 2*ones(1,K/2)];mu1 = ones(1,K);


Putting it to use - an examplePutting it to use - an example


resE = zeros(Np,K); resH = zeros(Np,K);

% compute time step sizexmin = min(abs(x(1,:)-x(2,:)));CFL=1.0; dt = CFL*xmin;Nsteps = ceil(FinalTime/dt); dt = FinalTime/Nsteps;


for INTRK = 1:5[rhsE, rhsH] = MaxwellRHS1D(E,H,eps,mu);

resE = rk4a(INTRK)*resE + dt*rhsE;resH = rk4a(INTRK)*resH + dt*rhsH;

E = E+rk4b(INTRK)*resE;H = H+rk4b(INTRK)*resH;

end% Increment timetime = time+dt;

endreturn

In the final routine, MaxwellDriver1D.m, the only significant di!erence fromAdvecDriver1D.m is the need to also specify the spatial distribution of ! andµ. In this particular case, we have assumed that these are constant in each el-ement but can jump between elements. Furthermore, we assume here that thejump is between the two middle elements. Clearly, this specification is problemdependent and can be changed as needed. Also, if ! and/or µ vary smoothlywithin the elements, this can be specified here with no further changes else-where.

MaxwellDriver1D.m

% Driver script for solving the 1D Maxwell’s equationsGlobals1D;

% Polynomial order used for approximationN = 6;

% Generate simple mesh[Nv, VX, K, EToV] = MeshGen1D(-2.0,2.0,80);


% Set up material parameterseps1 = [ones(1,K/2), 2*ones(1,K/2)];mu1 = ones(1,K);


epsilon = ones(Np,1)*eps1; mu = ones(Np,1)*mu1;

% Set initial conditionsE = sin(pi*x).*(x<0); H = zeros(Np,K);

% Solve ProblemFinalTime = 10;[E,H] = Maxwell1D(E,H,epsilon,mu,FinalTime);

To illustrate the performance of the developed algorithm, we considera problem consisting of a cavity, x ! ["1, 1], with metallic end walls [i.e.,E(±1, t) = 0]. Furthermore, we assume that x = 0 represents an interfacebetween the two halves of the cavity and that each half is a di!erent homoge-neous material. For simplicity we assume the materials are nonmagnetic (i.e.,µ(m) = 1,m = 1, 2).

This cavity supports a number of resonant modes of the form

E(m)(x, t) = "!A(m) exp(i!n(m)x) " B(m) exp("i!n(m)x)

"exp(i!t),

H(m)(x, t) =!A(m) exp(i!n(m)x) + B(m) exp("i!n(m)x)

"exp(i!t),

whereB(1) = exp("i2n(1)!)A(1), B(2) = " exp(i2n(2)!)A(2),

due to the boundary conditions,

A(1) =n(2) cos(n(2)!)n(1) cos(n(1)!)

, A(2) = exp(i!(n(1) + n(2))),

as a convenient normalization and ! is a solution to the equation

"n(2) tan(!n(1)) = n(1) tan(!n(2)),

found by requiring continuity of the field components across the materialinterface. Here and elsewhere above, the index of refraction is defined asn(m) =

#"(m).

For n(1) = n(2), this is a simple sinusoidal solution, whereas for n(1) $=n(2) the solution is more complex and loses smoothness across the materialinterface where the solution is only continuous.

We first solve the above problem with n(1) = 1.0 (i.e., vacuum), and n(2) =1.5, similar to glass, using di!erent numbers of elements, K, and order ofapproximation, N , in each element. All elements have the same size, h =2/K, and in the first case, we assume that K is even which guarantees thatan element interface is located at x = 0. On the left in Fig. 3.6 we showthe results, illustrating optimal convergence, O(hN+1), once the fields arereasonably resolved.



Both time-stepping and driver are essentiallyunchanged from the first case.


100 101 102 10310!710!610!510!410!310!210!1100101

K

||E!E

h||

h2

h5

N=1

N=2

N=3N=4

100 101 102 103

K

10!6

10!5

10!4

10!3

10!2

10!1

100

101

||E!E

h||

h

h2

N=1

N=2

N=3

N=4

Fig. 3.6. We show the global L2-error as a function of order of approximation, N ,and number of elements, K ! h!1, for the computed E-field solution to Maxwell’sequations in a metallic cavity with a material interface located at x = 0. On the left,the grid is conforming to the geometry; that is, there is an element boundary locatedat x = 0. On the right, this is violated and the material interface is located in themiddle of an element. The loss of optimal convergence highlights the importance ofusing a body-conforming discretization.

Also shown in Fig. 3.6 are the results for the case where K is odd, implyingthat x = 0 falls in the middle of an element. This shows a dramatic reductionin the accuracy of the scheme, which now is second order accurate. We alsoobserve that, for N even, the order is further reduced to first order. This canbe attributed to the fact that for N even, there is a grid point exactly atx = 0, whereas for N odd, x = 0 falls between two grid points. For N beingeven, one can redefine the material constant at x = 0 to be the average inorder to improve the convergence rate. A closer inspection of the result in Fig.3.6 for N = 3 indicates a less than second order convergence and a detailedanalysis reveals that the asymptotic rate should be O(h3/2) [234].

This latter example emphasizes the importance of using a geometry-conforming grid in which the elements align with features where the solutionloses smoothness in order to maintain the global high-order accuracy. How-ever, even if this is not possible, the results in Fig. 3.6 also show that althoughthe rate of convergence is reduced, the absolute error is lower for high-orderapproximations; that is, the constant in front of the error term decreases withincreasing order.

3.7 Exercises

1. Modify the solver for the scalar advection equation in Section 3.5 to solve

!u

!t+ a

!u

!x= 0, x ! [0, 2"],

Aligned grid Mis-aligned grid


Let’s summarizeWe have some the major pieces

‣ Some understanding of the key elements‣ Evidence of good behavior‣ Computationally well conditioned operators‣ Flexible and general implementations

However, many problems remain open

‣ What can we prove in terms of accuracy ?‣ How do we compute fluxes for systems ?‣ What about time-steps ?


Documents

DG-FEM for PDE’s Lecture 2Jan S Hesthaven Brown University [email protected] DG-FEM for PDE’s Lecture 2 Univ Coruna/USC/Univ Vigo 2010 Monday, June 7, 2010 A brief overview