Node Pairs Report

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High-resolution unstructured finite-volume

methods for conservation laws

A. Guar done 1 L. Quartapelle

1Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34,

20158 Milano, Italy. e-mail: [email protected]


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Abstract

The node-pairs representation of Galerkin and Finite-Volume methods for solving

conservation laws over unstructured multidimensional grids is described. This ap-

proach allows for the factorization of the grid-dependent metric quantities in the

computation of the finite-element integrals, thus leading to very efficient high-

resolution methods for simulating compressible flows in domains of arbitrary

shape. A procedure for imposing the boundary conditions in nonlinear hyper-

bolic systems within the assumed discretization framework is also presented. The

node-pair representation of second-order spatial derivative terms required to im-

plement LaxWendroff and TaylorGalerkin schemes is also described. The high-

resolution Galerkin andFinite-Volume formulationsarefirst developed for a scalar

conservation law and then extended to nonlinear systems of hyperbolic equations.Finally, an original TaylorGalerkin scheme for the mass conservation law over an

arbitrary velocity field is derived in both standard and node-pair formulations.


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Contents

1 Introduction 1

1.1 Scalar conservation law . . . . . . . . . . . . . . . . . . . . . . . 4

2 Finite element method 6


2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Spatially discrete form of the equations . . . . . . . . . . . . . . 13

2.4 Approximation of a diffusion term . . . . . . . . . . . . . . . . . 13

2.4.1 Basic property of diagonal stiffness elements . . . . . . . 15

3 Flux reinterpolation 17


3.2 Treatment of boundary conditions . . . . . . . . . . . . . . . . . 19

3.3 Spatially discrete form of the equations . . . . . . . . . . . . . . 21

4 Node-pair representation 23

4.1 Node-pairs and metric vectors of interaction . . . . . . . . . . . . 23

4.2 Proof of the split of domain and boundary contributions . . . . . . 25

4.2.1 Proof of the domain integral indentity . . . . . . . . . . . 25

4.2.2 Final transformation of the boundary term . . . . . . . . . 28

4.3 Node-pair form of the discrete equations . . . . . . . . . . . . . . 30

4.4 Treatment of boundary conditions . . . . . . . . . . . . . . . . . 31

4.4.1 Duplication and augmentation of boundary nodes . . . . . 314.4.2 Duplication of boundary edges in 3D problems . . . . . . 33

4.5 Diffusion term in node-pair form . . . . . . . . . . . . . . . . . . 35

5 Finite-Volume method on nonstructured meshes 38

5.1 Finite-Volume spatial discretization . . . . . . . . . . . . . . . . 38

5.2 Upwind Finite-Volume scheme . . . . . . . . . . . . . . . . . . . 40

5.3 The bridge between finite volumes and finite elements . . . . . . . 44

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6 Taylor-series-based integration schemes 47

6.1 Nonlinear scalar conservation law . . . . . . . . . . . . . . . . . 476.2 The fully discrete form of the equations . . . . . . . . . . . . . . 49

6.3 TaylorGalerkin scheme in node-pair form . . . . . . . . . . . . . 49

6.3.1 The bulk TG term . . . . . . . . . . . . . . . . . . . . . . 50

6.3.2 The boundary TG term . . . . . . . . . . . . . . . . . . . 52

6.3.3 Node-pair TaylorGalerkin scheme . . . . . . . . . . . . 54

6.4 Linear flux function . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 High-resolution scheme in node-pair form 59

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2 Extended node-pairs . . . . . . . . . . . . . . . . . . . . . . . . 617.3 High-resolution scheme for steady solutions . . . . . . . . . . . . 62

7.4 High-resolution scheme for unsteady solutions . . . . . . . . . . . 64

7.5 Limiters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8 Thermodynamics of gases 68

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.2 Definition of the gas properties . . . . . . . . . . . . . . . . . . . 69

8.3 Polytropic and nonpolytropic behaviour . . . . . . . . . . . . . . 74

8.4 Pressure function of the conservative variables . . . . . . . . . . . 78

8.5 Speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.6 Some particular gas models . . . . . . . . . . . . . . . . . . . . . 818.6.1 Polytropic ideal gas . . . . . . . . . . . . . . . . . . . . . 81

8.6.2 Nonpolytropic ideal gas . . . . . . . . . . . . . . . . . . 82

8.6.3 Polytropic van der Waals gas . . . . . . . . . . . . . . . . 82

8.6.4 Nonpolytropic van der Waals gas . . . . . . . . . . . . . 83

9 Euler equations of gasdynamics 85

9.1 Conservation laws of gasdynamics . . . . . . . . . . . . . . . . . 85

9.2 Euler equations in one dimension . . . . . . . . . . . . . . . . . . 86

9.2.1 Conservation variables . . . . . . . . . . . . . . . . . . . 86

9.2.2 Quasilinear form . . . . . . . . . . . . . . . . . . . . . . 879.2.3 Eigenstructure . . . . . . . . . . . . . . . . . . . . . . . 88

9.3 Euler equations in two dimensions . . . . . . . . . . . . . . . . . 89

10 Roe linearization of gasdynamic equations 92

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

10.2 Principles of Roe linearization . . . . . . . . . . . . . . . . . . . 93

10.2.1 Definition of Roe linearization . . . . . . . . . . . . . . . 93

10.2.2 General solution of Roe linearization . . . . . . . . . . . 95

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10.2.3 Solution in Jacobian form . . . . . . . . . . . . . . . . . 95

10.3 Linearization of Euler equations . . . . . . . . . . . . . . . . . . 9610.4 Solution of Euler linearization . . . . . . . . . . . . . . . . . . . 97

10.5 Polytropic van der Waals fluid . . . . . . . . . . . . . . . . . . . 102

10.6 Ideal gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

10.6.1 Flux functions homogeneous of degree one . . . . . . . . 107

10.6.2 Nonpolytropic ideal gas . . . . . . . . . . . . . . . . . . 108

10.6.3 Polytropic ideal gas . . . . . . . . . . . . . . . . . . . . . 110

11 High-resolution schemes for systems of conservation laws 113

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

11.2 Multidimensional upwind scheme for systems . . . . . . . . . . . 11411.3 Boundary conditions for system of conservation laws . . . . . . . 115

11.4 High-resolution scheme for steady solutions . . . . . . . . . . . . 118

11.5 A general limiter function for hyperbolic systems . . . . . . . . . 121

11.6 High-resolution scheme for unsteady solutions . . . . . . . . . . . 123

12 Boundary conditions in nonlinear hyperbolic systems 127

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

12.2 Conservative, characteristic and

physical variables . . . . . . . . . . . . . . . . . . . . . . . . . . 128

12.3 The boundary values for a scalar unknown . . . . . . . . . . . . . 130

12.4 Steps of the boundary procedure for a system . . . . . . . . . . . 13112.5 Alternative boundary procedure . . . . . . . . . . . . . . . . . . 134

A Discrete differential operators in node-pair weak form 140

B Nomenclature 142

C Algorithms 144

D Error analysis of the TaylorGalerkin method 152

D.1 Basic third-order TG scheme . . . . . . . . . . . . . . . . . . . . 152

D.2 Two-step third-order TG scheme . . . . . . . . . . . . . . . . . . 156

D.3 Two-step fourth-order TG schemes . . . . . . . . . . . . . . . . . 162

D.4 Vector advection equation . . . . . . . . . . . . . . . . . . . . . . 166

D.5 Mass conservation equation . . . . . . . . . . . . . . . . . . . . . 170


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If you think we are moving pictures, he said,You ought to pay you know. Moving pictures are

not made to be looked at for nothing. Nohow!

Robert Gilmore Alice in Quantumland


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Chapter 1

Introduction

The TaylorGalerkin scheme of Donea [3] has been proven successful for the

computation of unsteady hyperbolic problems and it has been used in a large

number of applications, ranging from multidimensional advection problems [5]

to the solution of the multidimensional shallow water equation [1]. Different

schemes have been proposed depending on the considered set of equations and on

theaimed-ataccuracy, but all rest on the same machineryfor the timediscretization

(by means of Taylor series) and for the space discretization (by means of finite

elements). As pointed out in [5], TaylorGalerkin schemes are to be considered

as the natural extension of LaxWendroff finite-difference schemes to the finite

element framework. In subsequent elaborations of the TaylorGalerkin approach,

Selmin [22] introduced a two-step procedure for time advancing. This lead to a

new class of schemes, in which, differently from the original ones, no modification

to the standard mass matrix is needed.

Although many applications of TaylorGalerkin schemes can be found in the

literature, the authors are not aware of any review work on the subject, the work of

different authors being mainlyfocusedon specializing the TaylorGalerkin scheme

to the problem under study. Moreover, the issue of the treatment of boundary

conditions for the hyperbolic problem in the TaylorGalerkin framework, in either

a strong or a weak sense, has received almost no attention in the literature.

A well known drawback of the TaylorGalerkin scheme lies in that it is builtupon finite element spaces of piecewise linear interpolations, so that this scheme

is not suitable for the computation of discontinuous or shocked solutions, as those

encountered, for instance, in the solution of the compressible Euler equations. To

overcome these difficulties, in [6] a two-step procedure that includes an artificial

dissipation operator was proposed. However, it is the authors opinion that this

issue is far from being clarified and deserves further attention.

The first part of the present work is intended as a review of the TaylorGalerkin

scheme for conservation laws; an error analysis of the scheme is also given in an

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appendix. For completeness, the description of the finite element and of Taylor

Galerkin schemes includes also an original procedure for the imposition of theproper (in flow or inlet) boundary conditions in a weak form.

TaylorGalerkin schemes are presented for both a multidimensional nonlinear

conservation law governing a scalar unknown and for a linear advection equation

(withvariablecoefficients)over a given advectionfield,possiblynonsolenoidaland

time-dependent, as it is thecaseof themass conservationequation forcompressible

flows. An original TaylorGalerkin scheme is presented for the mass conservation

law. This particular scheme can be applied in the solution of the Navier-Stokes

equations in those formulations where the mass conservation equation is tackled

is a separate step from the momentum and energy equations, to determine the fluid

density at a given time level. The new density so calculated is then employed toadvance themomentum and the energy variables, using an explicit timeintegration

scheme; such an approach is taylored to deal with the incomplete parabolic charac-

ter of the compressible NavierStokes equations. Moreover, the Taylor-Galerkin

scheme for linear advection can also be applied to the solution of the advection

problem arising from one or two-equation turbulent models.

In the second part of the report, we review the node-pair representation of

the Galerkin finite element method introduced by Selmin [23]. In particular a

node-pair formulation of the TaylorGalerkin schemes is derived as a step towards

their use for the solution of Euler and Navier-Stokes equations for compressible

flows. Thanks to the factorization of all the geometry-dependent quantities in thecomputation of the finite element integrals, a great improvement in the overall

efficiency of the code is achieved over the standard finite-element assembling

procedures. As a consequence, the proposed node-pair-based TaylorGalerkin

schemes opens theway to theuseof standard finite volume stabilization techniques,

such as upwind schemes, high-resolution schemes andartificial viscosity methods.

Interestingly enough, the node-pair representation of finite element schemes and

TaylorGalerkin schemes is discovered to provide a bridge between finite volume

and LaxWendroff schemes in the context of unstructured spatial discretizations.

This report is organized as follows. In Chapter 2 the finite element approxima-

tion ofa nonlinear scalar conservation lawis presented,and thediscrete formulation

of inlet boundary conditions for a hyperbolic problem is described. The finite ele-

ment approximation of a diffusionterm andof a second-order directional derivative

is detailed to construct TaylorGalerkin schemes for nonlinear conservation law.

In Chapter 3, an approximate techniquefor evaluating thenonlinear terms based on

the reinterpolation of fluxes, using the same local basis functions of the unknown

variable, is introduced. This step is preliminary to the derivation, in Chapter 4,

of the node-pair representation of the discrete equations. Chapter 5 introduces

the Finite-Volume method in the context of unstructured spatial discretizations.

The node-pair formulation of the upwind scheme is described together with the

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relationship between finite volumes and elements, as established in Selmins basic

work [22].Chapter 6 is devoted to the analysis of schemes based on the Taylor serie ex-

pansion in the time step, such as LaxWendroff and TaylorGalerkin schemes. In

particular the evaluation of the second-order directional derivative term is detailed

to build schemes of this kind for a nonlinear conservation law, both in the stan-

dard Galerkin form and in the node-pair representation. In Chapter 7 an original

implementation of high-resolution and TaylorGalerkin schemes for discontinu-

ous solutions over unstructured meshes is presented for a scalar conservation law.

Before describing the extension of the high resolution unstructured method to

the nonlinear hyperbolic system of the gasdynamic equations, some preparatory

material is introduced.Chapter 8 presents the thermodynamical properties of gases and describes the

gas models that will considered in the subsequent applications. The Euler equa-

tions of gasdynamics governing the motionof a compressible but invisicd fluid are

recalled in Chapter 9 for both one- and multidimensional flows; the eigenstructre

of these systems of nonlinear hyperbolic equations is derived. Chapter 10 presents

the idea of local linearization introducedby Roe to replace the Riemann problem at

each interface by an approximate linear problem that is equivalent from theconser-

vation viewpoint to the original nonlinear problem. In particular, the linearization

is sought for by determining an intermediate state such that the Jacobian matrix

satisfies Roes conservation conditions. The solution of the linearization problemis provided for Euler equations made complete by some ideal and nonideal gas

models.

All these physical and mathematical components are needed to develop the

high-resolution schemes for solving the multidimensional Euler equations which

are detailed in Chapter 11. The last Chapter provides a detailed account of the

original procedure for satisfying the boundary conditions in the simple case of

one-dimensional hyperbolic systems, whose multidimensional counterpart was

employed in the preceding chapter.

In theappendices, thenode-pair representationof differentialoperators in weak

form is summarized (Appendix A) and a short table of the nomenclature is given

(Appendix B). In Appendix C, an algorithm for the construction of the divergence

operator in node-pair format is detailed and, in Appendix D, the two-step Taylor

Galerkin schemes are recalled and an error analysis of these schemes is provided.

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1.1 Scalar conservation law

We start byconsidering a multidimensional conservation lawfor thescalarquantity

u, written in the following divergence form

u

t+ f(u) = 0, (1.1)

where the flux f(u) Rd is a given vector function of the unknown u(x, t) R,x Rd, t [0, T] (see figure 1.1). Here, d = 2 or d = 3.

Figure 1.1: The domain and its boundary in two spatial dimensions.

The conservation law under consideration must be supplemented by proper initial

and inlet boundary conditions, so that the complete initial boundary value problem

(IBVP) reads (e.g. [20])

u

t+ f(u) = 0,

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

|in

=a(s, t), s

in,

(1.2)

where u0(x) and a(s, t) are known functions, defined their respective domain

and in. The coordinate s over the boundary is indicated here as a scalar

variable with reference to problems in two dimensions, while it becomes a vector

(of dimension two) in the three-dimensional case. The inflow or inlet portion

in of the domain boundary is defined as (figure 1.2)

indef= x n(x) a(u(x, t)) < 0

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a

out

in

Figure 1.2: The (local) direction of the advection field a with respect to the outwardnormal n identifies the inflow and outflow portions of the boundary.

with n(x) denoting the outward normal unit vector and with the advection velocity

defined by

a(u)def= d f(u)

du.

The remaining part of the boundary will be referred to as the ouflow portion of

and will be denoted by out, so that

= in

out. Note that the

definition of in is a function of the trace of the solution u(x, t) at a given time,so the inflow boundary is in general dependent on time and a more precise notation

for it would be tin, and similarly for tout

For a complete analysis of the well-posedness of the IBVP (1.2) including the

compatibility conditions on the data we refer to the monograph of Godlewski and

Raviart [9], chapter 5.

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Chapter 2

Finite element method

In this chapter, the scalar conservation law (1.2) is discretized in space by means

of the standard finite element method (FEM). For later convenience, the FEM

approximation of the boundary terms is detailed, together with a procedure to

impose inlet boundary conditions in this framework.

Figure 2.1: Triangulation of the domain .


The scalar conservation law (1.1) is recast in a weak or variational form by

applying the classical Galerkin finite element method. The weak form of the

equation is obtained (e.g. [20]) by multiplying the differential equation by test

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functions belonging to a suitable1 space V

H1(), and integrating over the

domain as follows ,

u

t

+ , f(u) = 0,

where (, ) denotes the inner product in L2(). To simplify the presentation, thediscussion of how the boundary conditions are imposed is postponed to Chapter

2.2. An integration by parts gives

,

u

t

, f(u)

+

n f(u) = 0. (2.1)

Let us now moveto the discrete form of the equations above; if a finitedimensional

space Vh H1(), of weighting functions h of linear or bilinear Lagrangiantype is considered (figure 2.1) , we obtain

i

iu

t

i

(i ) f(u) +

i

i n f(u) = 0, (2.2)

where the shorthand notation i i has been introduced. In writingthe boundary integral, we have taken into account that the element basis function

i

Vh of each node i vanishes on the boundary i of its support i , but for

nodes on the domain boundary. The support i is the union of the sets e, edenoting the e-th finite element, which contain the node i (cf. figure 2.2 and 2.3).

i

i

i

Figure 2.2: The shape function i (x) in one spatial dimension and its support i .

1The introduction of the Sobolevspace H1( ) in the context of hyperbolicproblems cannot be

justified mathematically unless thesolution has a regularity typical of elliptic or parabolic (second-

order) problems.

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i

Figure 2.3: The support i of the shape function i (x) in two spatial dimensions.

Let us assume that the solution u(x, t) is approximated by the function uh (x, t)

obtained by an expansion in the same space of the weighting functions h(x) as

follows,

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

kNuk(t) k(x), (2.3)

where uk(t) is the value of the approximate solution at node kand at time t (figure

2.4) and N denotes the set of all nodes of the triangulation. The weak form of the

conservation law now readsi

iuh

t

i

(i ) f(uh ) +

i

i n f(uh ) = 0.

u

uh

Figure 2.4: The function uh(x, t) versus u(x, t) in one spatial dimension for linear La-grangian finite elements.

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Substituting the expansion ofuh into the first term and transferring the other two

terms into the right-hand side giveskNi

ik

i k

duk

dt=

i

(i ) f(uh )

i

i n f(uh ).

By recalling the definition of the elements of the consistent mass matrix, namely,

Mi k =

ik

i k, (2.4)

the (Bubnov) Galerkin finite element approximation of the scalar conservation

law assumes thekNi

Mi kduk

dt=

i

(i ) f(uh)

i

i n f(uh). (2.5)

2.2 Boundary conditions

Let us recall thenonlinear conservation-law equation endowedwith inlet boundary

condition as in (1.2), namely,

u

t + f(u)

=0,

u|in = a(s, t) s in.Under this inlet condition the boundary integral of (2.1) can be split in the sum of

two integrals, as follows,

n f(u) =

out

n f(u) +

in

n f(a), (2.6)

where outdef= \ in and where the integrand function over the inlet part

in of the domain boundary has been evaluated by taking into account the pre-

scribed boundary condition u|in = a(s, t). We remind that the partition of theboundary in in and out depends on the advection field a and hence, in

the nonlinear case of interest here, the partition depends on the boundary value of

the unknown u, so that the inflow and outflow portions of may change with

time and ought to be denoted more appropriately as

tin and tout.

The type of boundary condition changes along the boundary and the points of

transition, where the sign ofn a changes, may fall, in general, inside a boundary

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edge. This represents the most difficult occurrence, in which case the imposition

of the boundary condition for the hyperbolic equation is particularly critical andrequires an ad hoc special treatment, see below. By contrast, when the change of

the boundary condition type from inflow to ouflow occurs across two consecutive

edges, the account of the boundary condition less complicated.

The evaluation of the boundary integrals (2.6) of function n f(u) over the

solution-dependent domains in and out in the context of the finite element

method can be performed as follows. Let us introduce the finite element approxi-

mation of the unknown u and of the function a defined on the boundary,

u(x, t)| out

uouth (s, t)def

= kNout uk(t) k(s), s

out

a(s, t) ah (s, t) def=

kNinak(t)

k(s), s in

where k is the trace of the weight function k on , namely,

k(s)def= k(x(s)), x(s) ,

and Nout and Nin are the boundary nodes belonging to the outflow and inflow

portions of , respectively. The finite element approximation of the boundaryintegral (2.6) now readsout

i n f(u) +

in

i n f(a)

out

i n f(uouth ) +

in

i n f(ah ).

For completeness, we describe the numerical computation of the above integral by

means of approximate quadrature formulas. If a Gaussian numerical integration

is considered, the distinction between in and out pertains only to the Gauss

points, where the integrand is evaluated. The two contributions to the boundary

integral will be computed as followsout

i n f(u) +

in

i n f(a)

gGouti (xg)ng f

k uk

k(xg)

pg

+

gGini (xg)ng f

k ak

k(xg)

pg,

where the subscript g is used to denote Gauss integration points and pg is the

corresponding integration weight. The two sets of Gauss integration points Gout

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Grid nodeGauss point

a

a

Gout

Gout

Gout

Gin

Gin

Gin

Figure 2.5: Gauss points (indicated with empty circles) belonging to Gout and Gin.

and Gin ofout and in, respectively, are defined as follows (figure 2.5)

Goutdef= g G ng ak uk k(xg) 0

Gindef= g G ng ak uk k(xg) < 0

In other words, at each Gauss integration point the (interpolated) value of the

unknown is used to evaluate the advection field, and, consequently, to determine

whether an inlet boundary condition has to be imposed or not.

The approximate integration procedure outlined above stems directly from

equation (2.6), in that the resulting algorithm retrieves first the partitions outand in and then performs the integration. This procedure is computationally

inefficient, since it can be either timeconsuming (the isoparametric transformation

of the whole boundary is to be performed twice) or memory consuming (the coor-

dinate of Gauss point and the value of the shape functions are to be stored for the

whole boundary at thesame time). In practice, since in theusual implementationof

the finiteelement method the computation of the integrals is performed by splitting

them into a sum of elemental contributions, the above procedure can be applied

on every element independently, without loss of efficiency. In the following, we

will write the boundary integrals in an alternative but equivalent form that is more

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suitable for descriptive purposes and also to deal with different formulation of the

finite element method.First of all, we want to encompass the different definitions of the integral

over out and in within a uniform procedure. This can be accomplished by

introducing the function

u(s)def=

u if n(s) a(u(s)) 0a if n(s) a(u(s)) < 0 (2.7)where the binary operator | indicates the possible choice between u and a, depend-ing on the local direction of the advection field a(u) with respect to the outward

normal n(s), s , as shown in figure 2.6. It follows that the boundary term

u

a

u(s)

n a

Figure 2.6: The function u(s) in two spatial dimensions.

(2.6) of the weak equation can be written in the compact form

out n f(u) +

in n f(a) =

n f(u).

Coming now to the discrete representation of the unknown and of the prescribed

boundary value we have

n f(u)

i

i n f(uh (s)). (2.8)

Thus, in this formulation, the subdivision of the boundary into the sets inand out is obtained automatically through evaluating the integral of a function

of the unknown uh or of the datum ah over the whole boundary .

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2.3 Spatially discrete form of the equations

For completeness, we summarize here the detailed form of the spatially discrete

counterpart of the conservation law (1.1) according to the standard finite element

approach. We havekNi

Mikduk

dt=

i

(i ) f(uh )

i

i n f(uh(s)), (2.9)

where we have introduced the following (semi-) discrete approximation of the

unknown

u(x, t) uh (x, t) = kNi uk(t) k(x).The function u is defined as

uh (s)def=

uh if n(s) a(uh ) 0ah if n(s) a(uh ) < 0in which the following expansion of the boundary datum has been considered

a(s, t) ah (s, t) = kNiak(t)

k(s), s in.

As well known [7], the finite element method described above suffers from a

deterioration of accuracy for increasing t in hyperbolic and advection problems.

In chapter 6, we shall recall the TaylorGalerkin scheme of Donea [3] which has

been devised to overcome these difficulties for the particular case of transient

solutions.

2.4 Approximation of a diffusion term

As a preliminary step in the derivation of the TG scheme for the scalar conservationlaw, the spatial discretization of a diffusion term is presented. Let us consider the

problem of approximating a term of the type

(u) (uh )

by means of the finite element method, where u is a scalar unknown, for example

the temperature or a velocity component, and is a (possibly variable) coefficient,

for example a thermal conductivity or a viscosity coefficient. Once expressed in

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weak form byprojection onto a finite dimensional space Vh

H1() of weighting

functions h Vh of Lagrangian type, the viscous term integrated by parts readsi

i (uh ) =

i

(i ) uh +

i

i n uh . (2.10)

The evaluation of the boundary term in the expression above depends on the kind

of boundary conditions specified for u, as a consequence of the elliptic nature of

the term under examination: thus Dirichlet or Neumann boundary conditions are

usually prescribed on different parts of. Suppose that these conditions are

u

=a on D and

(u/n) n uh = b on N,(2.11)

where D N = . Then, the weighting functions are to be chosen in thespace H1D(), namely, such that i = 0 on D (see figure 2.7).

Dirichletboundary condition

Neumannboundary condition

xx0 x1

1

0

Figure 2.7: Shape functions belonging to the Sobolev space H1D([x0,x1]) in one spatialdimension. A Dirichlet boundary condition is to be imposed at x = x0, while a Neumannboundary condition is assumed at x = x1.

As a consequence, expression (2.10) becomesi

i (uh ) =

kNi

ik

(i ) k

uk +

iNi b. (2.12)

Moreover, if the viscositycoefficient is constant, thediffusionterm can be written

in the form i

i (uh ) =

kNiKik uk +

i N

i b (2.13)

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where the elements Ki k of the stiffness matrix K are defined as follows

Ki kdef= i ,k =

ik

(i ) k (i = k), (2.14)

which is symmetric. Here, the usual symbol K for the stiffness matrix has been

retained, which should not be confused with the node index k or the calligraphic

letter N denoting node sets.

In conclusion, if an expansion of the Neumann boundary term into the finite

dimensional space of weighting function is considered, i.e.,

b(s, t)

kNN bk(t) k(s), s

N

where k is the trace of the k-th shape function k on defined in (2.2), and

NN is the set of the Neumann boundary nodes, we have, for a constant dissipative

coefficient ,i

i (U) =

kNiKik uk +

k(Ni NN)

ik

i k

bk,

where the shorthand notation

i k i

k

has been introduced.

Instead, for nonconstant , the discrete representation of the Laplacian term under

consideration readsi

i (uh) =

kNiKi k uk +

k(Ni NN)

ik

i k

bk, (2.15)

where we have introduced the more general stiffness matrix

Kikdef=

i , k

=

ik (i ) k. (2.16)

2.4.1 Fundamental property of diagonal elements

of the stiffness matrix

The diagonal elements of the stiffness matrix have an important property that

will prove very useful in the explicit matrix multiplication within the edge-based

organization of thespatiallydiscrete data that will be introduced in thenext chapter.

The diagonal element of each row of the stiffness matrix is the opposite of the sum

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of all off-diagonal elements of the considered row, namely, the following relation

holdsKi i =

kNi,=

Kik. (2.17)

This identity is a simple consequence of the fact that the stiffness matrix is the

discrete representation of the (negative) Laplacian for the Neumann problem and

that the solution to this problem is defined up to an arbitrary additive constant.

This arbitrariness is possible in the discrete problem only provided that the sum of

all elements of each raw of the matrix vanishes.

The detailed proof the identity follows from the local support property of the

shape functions and the basic relationkNei

k(x) = 1, x e, e E,

over any finite element e of the triangulation. This relation implies that

i (x) = 1

kNei , k=ik(x) x e,

and hence

i (x)

= kNei , k=i k(x) x e.

From the definition of the diagonal element we have

Ki idef=

i

|i |2 =

eNei

e

|i |2.

Using the relation for the gradient just established, we have

Ki i =

eNei

e(i )

kNei , k=ik

=

eNei

kNei , k=i

e

(i ) k

=

kNi , k=i

ik

(i ) k =

kNi,=Kik.

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Chapter 3

Flux reinterpolation

In the following sections, we shall introduce an approximation of the flux function

which makes the evaluation of the corresponding discrete terms much simpler and

more efficient than in theclassical Galerkin formulation. Suchan approximation is

based on reinterpolating the flux function f(uh ) using the flux values calculated at

the nodes and the same basis functions employed in the expansion of the unknown.

Theflux reinterpolationhas been frequently considered in the solution of problems

by means of the finite element method and it is known as group-representation

method or group approximation or under other names; see, for example, [18].

The interpolation of the flux is actually an exact representation in the particular

case of a linear flux function f(u) = a u, with a constant; on the contrary, for agiven arbitrary f(u) the reinterpolation introduces numericalerrors whose analysis

is beyond the scope of this work. Nevertheless, it would be worth investigating the

effects of using such an approximation especially in the case of Lagrangian finite

elements of order higher than one.


Let us first examine the volume integral of (2.5) obtained by the integration by

parts of the divergence f(u) of the flux function, namely,i

i f(uh) =

i

i f

kN ukk

(3.1)

which constitutes the approximation of the corresponding integral under the con-

sidered Galerkin approach. We now reinterpolate the flux by means of the same

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i ki k

Figure 3.1: The intersection ik of the supports i and k of nodes i and k.

basis functions h , i.e., we make the approximation

f(uh (x, t)) = f

kN uk(t)k(x)

kN

f(uk(t)) k(x) =kN

fk(t) k(x)

where fk(t)def

= f(uk(t)). Thus, the spatial integral (3.1) is approximated byi

f(uh ) i

i

i (x) kN

fk(t) k(x) =

kNifk(t)

i

k(x)i (x)

and also, denoting more precisely the actual integration domain (cf. figure 3.1),i

i f(uh )

kNifk

ik

ki (3.2)

In a similar way, let us now evaluate the boundary term provided by the integration

by parts. Resorting to the same reinterpolation of the fluxes, we have i

i n f(uh )

i

ikN

n f(uk) k =

i

i

kNin f(uk) k

=

kNifk

ik

i k n

where, again, fk = f(uk). The boundary integral will be different from zero onlyprovided that also the node k belongs to , so that, to indicate this, the range of

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the summation index in the previous relation is restricted appropriately as followsi

i n f(uh )

kNifk

ik

i k n, (3.3)

where N represents the set of all boundary nodes.

In conclusions, once the approximation based on the reinterpolation of the flux

has been introduced, the discrete Galerkin formulation of the conservation law

reads

kNi Mikdu k

dt = kNi fk ikki kNi fk iki k n. (3.4)The introduction of the flux reinterpolation has allowed us to recast the integrals

in space involving the (time-dependent) flux function as the product of the time-

dependent nodal value fk(t) and fixed spatial integrals involving the shape func-

tions i and k. Therefore, the computationof the integrals dependent on the mesh

geometry appearing in the discrete equation can be performed once and for all at

the beginning of the computation, in a preprocessing phase. This leads to a strong

reduction in the CPU time requirements with respect to the standard Galerkin ap-

proach (cf. equation (2.5)). On the other hand, such an approximation introduces

an integration error whose analysis is beyond the scope of this work.

3.2 Treatment of boundary conditions

To simplify thederivationof expression(3.4) wedidnotconsider theproperbound-

aryconditionsfortheinitial boundaryvalueproblem (1.2). Moreover, in thederiva-

tion of (3.4), and in particular in (3.3), we assumed a constant flux function over

the intersection of the twosupport i and k of the nodes i and k. Wheneverthe

type of the boundary condition changes on a boundary element edge, as described

in Section 2.2, this approximation is no longer valid and the variable f cannot be

factorized anymore. In the following we exclude this occurrence and discuss thetheme of the imposition of the boundary conditions assuming that they can change

type only when passing across different boundary elements, as depicted in 3.2.

In this section, we address the problem of imposing the proper inlet boundary con-

ditions, and, more generally, of imposing boundary condition of different type in

this framework. Under the flux reinterpolation already introduced, we seek for

a suitable form of the boundary term which preserves the envisaged factorization

of the geometry-dependent part from the time-dependent one. The resulting for-

mulation is obtained without introducing any approximation other than the usual

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aa

a

a

out

in

i km

j

n

n

n

Figure 3.2: Over the boundary k of the support k of node k boundary conditions ofdifferent kind are to be imposed: on k j an inlet boundary condition is required,while k i is an outflow region, that is, no boundary conditions are to be imposed.

group representation of the fluxes in the case of linear and bilinear Lagrangian

elements, and a first order approximation of the normal direction in the case of

elements of order p > 1.We perform the following reinterpolation of the flux function on the boundary

on which we impose

f(s, t)

kNfk(t)

k(s), (3.5)

where

fk(t)def= fu(sk, t). (3.6)

In the last expression, function u(s, t) is defined as in (2.7), namely:

u(s, t)def

= u(s, t) if n(s) a(u(s, t)) 0a(s, t) if n(s) a(u(s, t)) < 0 (3.7)where u(s, t) = u(x, t)| , that is the trace of the solution u on the boundary attime t. [Note that a is the boundary value prescribed on u and is not the intensity

of the advection velocity a.]

In the reinterpolation, the dependence of f(s, t) on the boundary coordinate s

cannot be dropped, as done for instance for the coefficient fk(t) of the reinterpo-

lation in the domain, for u here depends on s. By substituting (3.3) in (2.8), we

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have i

i (s)n f(s, t) kNi

i

i (s) k(s)n(s) fk(t).

In the caseof linear or bilinear elements, the normaln is constant over theboundary

e belonging to the element e, so that we can define the outward normal of

element e as

nedef= n(s) = constant s e (3.8)

and the following restriction of f(u(s, t)) over the e-th element as (figure 3.3)

fe

k(t)def= f(u(sk, t)) s ek , (3.9)

which is a function of the time t only and can be factorized in the computation ofthe space integrals, as

i

i n f(u) eEi

kNi

fe

k(t)

ike

i k n. (3.10)

outout in

i km j

n a n a

f(um )

f(ui )

f(uk)

f(ak)

f(aj )

Figure 3.3: Flux reinterpolation at boundaries for the situation sketched in figure 3.2.

3.3 Spatially discrete form of the equations

For completeness, let us recall here the complete form of the spatially discretized

conservation law (1.1) under the hypothesis of flux reinterpolation. We havekNi

Mikduk

dt=

kNi

fk

ik

ki +eEi

kNi

fe

k

ike

i k n, (3.11)

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where we have defined as usual

fk(t)def= f(uk(t)) (3.12)

and

fe

k(t)def= fu(sk, t), s i k e (3.13)

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Chapter 4

Node-pair representation

In this chapter we introduce a more convenient way of writing the flux terms

of the discrete equations which has been proposed by Vittorio Selmin [23]. In

his formulation, these contributions are expressed as a summation over pairs of

interacting nodes in such a way quantities dependent on time are factorized out

from quantities dependent on the assumed spatial discretization. As a results, all

the quantities associated with geometrical features of the mesh can be evaluated

in a preprocessing phase of the computation. Moreover, the flux contributions are

found to be expressed in a quasi one-dimensional form that permits theexploitation

of all algorithmic tools developed for the solution of one-dimensional conservation

law equations and systems, as, for example, upwind schemes.

4.1 Node-pairs and metric vectors of interaction

Let us introduce the following metric vector quantities

ikdef=

ik

(ik ki ), (i = k) (4.1)

which are different from zero only if i k = 0, i.e., if the nodes i and kinteracts in the finite element sense. Each couple of such interacting nodes

is called node-pair (figure 4.1). In terms of these metric vectors, the discrete

counterpart of the considered scalar conservation law can be recast in the compact

form [23]kNi

Mikduk

dt=

kNi,=

fk + fi2

ik

+

kNi,=

fk fi2

ik

i k n

kNi,=fk

ik

i k n

(4.2)

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i k

Figure 4.1: The node-pair i-k.

where the last two summations must be noticed to involve only boundary nodes.

Themetric vectors iks occurring in equation (4.2) above satisfy the following

fundamental properties [25]:

ik = ki , (antisymmetry) (4.3a)ii

=0 (4.3b)

kNiik = 0, for any internal point i (4.3c)

kNi

ik =

i

i n =

i

i n (4.3d)

Properties (4.3a) and (4.3b) are important for ensuring a conservative numerical

scheme, while properties (4.3c) and (4.3d) are a necessary consequence of the

fact that a constant flux must give a zero contribution to the change in uh at each

internal point. The properties listed above allow to identify the quantities ikas metric vectors which behave similarly to the integrated normal vectors of a

control volume in the finite volume approach. From this viewpoint, properties

(4.3c) and (4.3d) guarantees that the control volume is closed. In Selmin [23], an

equivalence theorem is given which relates the node-pair formulation of the finite

element method to the standard finite volume approach over suitable constructed

control volumes on triangular meshes. The equivalence holds for the discrete

equations associated with internal nodes, but it breaks down for those associated

with boundary nodes.

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4.2 Proof of the split of domain and

boundary contributions

Equation (4.2) is proved starting form the discrete equation (3.4), which is repeated

here for convenience:kNi

Mi kduk

dt=

kNifk

ik

ki

kNifk

ik

i k n, (4.4)

and using the following identity

kNi fk ikki = kNi,=fk

+fi

2

ik + kNi,=

fk

fi

2 i i k n

(4.5)

which splits the domain integral into domain contributions and boundary contribu-

tions. Thedirect substitutionof (4.5) intoequation (4.4) gives (4.2). Theboundary

contributions in (4.2) are recast in a different more convenient form by means of a

second identity that will be discussed below.

4.2.1 Proof of the domain integral indentity

The standard method used in finite element schemes for computing the integral

on the left hand side of the relation above is to assemble the contributions coming

from each element e in the mesh, exploiting the local support property of the shape

functions, as followskNi

fk

ik

ki =eEi

kNe

fk

e

ki , (4.6)

where Ei is the set of the elements having the node i in common (bubble around

node i) and Ne is the set of the nodes of element e. The first summation on the

right-hand side is limited to the elements contained in the support i of node i ,

which are the only ones to give a nonzero contribution to integrals containing the

function i . In fact, if we indicate by e the subset of pertaining to the e-thelement ( = eE e) we have, by definition i = eEi e. Let us considerthe identity (gradient theorem)

e(i k) =

e

i k n (4.7)

valid for any continuous function i k, which gives the following relatione

ki =

eik +

e

i k n (4.8)

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i k

e1

e2

e1i k

e2i k

ik

Figure 4.2: Elemental contributions to the metric vector ik.

By virtue of this identity one can deducese

ki =1

2

e

ki +1

2

e

ki

=1

2eki + 12 eik + ei k n=

ei k

2+ 1

2

e

i k n (4.9)

where in the last equality we have introduced the elemental contributions ei k to

the metric vectors i k, according to the definition (figure 4.2)

ei kdef=

ike(ik ki )

In terms of these elemental contributions the metric vector associated with the pairi -k can be expressed as

ik =

e(EiEk)ei k

Using (4.9), the integral (4.6) becomeskNi

fk

ik

ki = eEi

kNe

fk

eik2

12

e

i k n

(4.10)

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On the other hand, from equation (4.9) it follows that

ei k = 2

eki +

e

i k n

and also, recalling (4.8),

ei k = 2

eik

e

i k n

Summing up this relation for all nodes k belonging to element e and using the

elementary property

kNe k(x) = 0, x e, we obtain

kNe eik + ei k n = 0

Summing up this relation for all elements belonging to Ei and multiplying the total

by the constant vector fi , we obtain the following relationeEi

kNe

fi

ei k +

e

i k n

= 0

Multiplying this relation by 1/2 and adding it to the right hand side of (4.10) we

obtain

kNi

fk ik

ki = eEi

kNe

fk + fi2

ei k+eEi

kNe

fk fi2

e

i k n.

By recalling that eik = 0 for e (Ei Ek) and ik =

e(EiEk) ei k and that

i i = 0, we can rearrange the right-hand side of the last relation so as to obtainkNi

fk

ik

ki =

kNi,=

fk + fi2

ik +

kNi,=

fk fi2

i

i k n,

where Ni,= denotes, as already specified, all mesh points belonging to i exceptfor the node i itself. Since i = 0 on i unless the node i belongs to , thelast term can be written as

kNi,=

fk fi2

ik

i k n

The boundary integral will be different from zero only provided that also the node

k belongs to , so that, to indicate this explicitly, the index of the summation is

rewritten as follows kNi,=

fk fi2

ik

i k n

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Therefore the final result iskNi

fk

ik

ki =

kNi,=

fk + fi2

ik +

kNi,=

fk fi2

ik

i k n,

that is, the identity (4.5).

Exploiting the identity (4.5) just established, the starting equation (4.4) reduces

to equation (4.2) that will be written compactly askNi

Mikdu k

dt=

kNi,=

fk + fi2

ik + BTi , (4.11)

after having introduced the following notation for indicating together the two

boundary contribution terms

BTi =

kNi,=

fk fi2

ik

i k n

kNi,=

fk

ik

i k n. (4.12)

4.2.2 Final transformation of the boundary term

The two integrals contributing to expression (4.12) can be recast in an alternative

manner that involves theuseof metricvectors associated with node-pairs belonging

to the boundary as well as with the nodes of the boundary. The final form givesthe boundary terms where the boundary conditions can and must be enforced in

the Galerkin FE approache, in a weak form.

Initially, let us rewrite the second term (apart from its sign) of BTi by isolating

the contribution of the node i , as followskN

i

fk

ik

i k n =

kNi,=

fk

ik

i k n + fi

ik

i i n

We now add and subtract the term

kNi,=

fi

ik

i k n

to the relation above to obtainkNi

fk

ik

i k n

=

kNi,=( fk fi )

ik

i k n + fi

kNi

ik

i k n.

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Byexploiting thelocal support propertyof theshapefunctions and thebasicrelationkNe

k(x) = 1, x e, e E,

we finally obtainkN

i

fk

ik

i k n =

kNi,=

( fk fi )

ik

i k n + fi

i

i n

By substituting this result into (4.12) we note that there are two equal terms but

with coefficients 1

2

and

1, so that the boundary contribution to the i -th discrete

equation assumes the following from:

BTi =

kNi,=

fk fi2

ik

i k n fi

i

i n, (4.13)

where the first integral is over the domain i k = ik , and therefore itexists only for i, k , with however i = ksince for i = kthe vector coefficientvanishes.

We are therefore led to introduce the following boundary metric vectors

i kdef=

ik

i k n, i, k N (i = k)

idef=

i

i n, i N

in terms of which the boundary contribution is written in the final form

BTi =

kNi,=

fk fi2

i k fi i . (4.14)

We note thatik is definedonly for i, k N andunder thecondition i k =. Moreover, i is similarly defined only for i N .

The symmetry property ik = ki can be contrasted with the antisymmetryproperty i k = ki . However, the coefficients in front of the two metric vectorsare respectively symmetricandantisymmetric, so that thetwocorresponding scalar

products share one and the same property of antisymmetry with respect to the

indices i and k. Stated in other words, the contribution of all node-pairs, both the

domain terms and the boundary ones, repects conservation exactly.

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4.3 Node-pair form of the discrete equations

Let us now recall the conservation law (1.1) we started from, i.e.,

u

t+ f(u) = 0.

Once the metric vectors have been introduced, the complete final form of the

spatially discrete conservation law in the node-pair representation readskNi

Mi kduk

dt=

kNi,=

ik fi + fk

2

kNi,

=

i k fk fi

2 i fi ,(4.15)

where fidef= f(ui ). The last, nodal, contribution i fi is present only when node

i lies on the boundary : in other words, when i correponds to an internal point,

the respective equation does not contain any term of this form. This circumstance

is reminded by the superscript index appended to boundary metric vector i .

To summarize all elements of the node-pair formulation, we collect here the

definition of the mass matrix M and of the metric vectors , and , namely

Mik =

ik

i k,

ik =

ik

(ik ki ), (i = k)

ik =

ik

i k n, i, k N (i = k)

i =

i

i n, i N

Remarkably, all the geometrical information pertaining to the triangulation are

contained in the metric quantities above.

Furthermore, the evaluation of the summations in the right-hand side of the

discrete equations can be performed quite easily by processing sequentially the

list of node-pairs and accumulating the left and right contributions with the

proper sign according to their overall antisymmetry. More precisely, the list of

all interacting node-pairs involved by the domain contribution term is stored in a

suitable array whose entries contain the order numbers of two nodes of each node-

pair. This array contains therefore the connectivity information of the entire mesh

seen from the viewpoint of knowing the interaction between the nodes as brought

about by the overlapping of the their respective support. By means of this nod-pair

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connectivity, the summation of the terms on the right-had side can be performed

explicitly by accounting for thecontributionof eachnode-pair term to thequantitiesof two nodes. A similar array for node-pair-to-nodes connectivity is memorized

for the set of node-pairs involved by the summation associated with the boundary.

The corresponding boundary connectivity information allows to account for the

boundary integral expressed in the node-pair format, similarly to what has been

explained for thedomain integral. Finally, thecontribution of thethird term is taken

into account by looping on the list of all boundary nodes. The latter must therefore

been memorized as another boundary information for the considered mesh. Thus,

the node-pair representation of the spatial triangulation comprises three differents

pieces of information for the mesh topology, one pertaining to the domain and

the other two associated with the boundary, while a FEM mesh is described byonly two connectivity element-to-nodes arrays, one for the domain elements and

the other for the boundary element. The resulting algorithm for performing the

explicit accumulation of the right-hand side in node-pair format is straight forward

and an example for a conservation law is given in appendix C.

4.4 Treatment of boundary conditions

For simplicity, the node-pair form of the spatially discrete equations has been de-

rived in the previous sections without considering the imposition of the boundary

conditions of the considered hyperbolic problem. In particular, the boundary inte-gral has been rearranged by assuming that a constant reinterpolated flux function

fi over the support i ofi . This assumption cannot be accepted when boundary

conditions change from inflow to outflow along the same element, as pointed out in

section 3.2. As a consequence, the envisaged factorization of the flux function in

theboundary integrals is notallowed and theformulation must be modified slightly

so as to include the possibility of imposing boundary conditions of different type.

Let us recall the form of the boundary integral which has been obtained under

the hypothesis of flux reinterpolation. We have

i n f(u) eEi

kNi

f

e

k ik ei k n (4.16)where

fe

k(t)def= fu(sk, t), s e (4.17)

4.4.1 Duplication and augmentation of boundary nodes

To simplify the analysis, let us assume that the partition of the boundary in its

inflow and outflow parts does not occurs inside any boundary elements. In

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other words, we are assuming that the frontier between these two parts is precisely

located at points (in 2D) and along edges (in 3D) which are on the border of theboundary elements.

Under this assumption each boundary element can be associated either to the

inflow boundary or to the outflow boundary, according to the sign of n f,

where f is evaluated in the interior of the boundary element. This circumstance

allowstosplitthenodalterm i fi in itscontributionsdueto thedifferentboundary

elements. In 2D problems, there are always two distinct contributions, stemming

from the two boundary edges containing the boundary node i . Therefore one can

augment artificially the number of boundary nodes, by duplicating all of them,

so as to compute the term of the purely nodal contribution still as a single cycle

running on a double number of artificially augmented boundary node.On the contrary, in 3D problems there a few of boundary element containing

the boundary node i as one of their vertices, the typical number being around 5 or

6. In this case, it is necessary to known the number Ni of these boundary triangles

associated with theboundary node i . Then onecan augment artificially the number

of boundary nodes by replicating Ni times the surface node i .

By summarizing, while the numbering of domain nodes is left unchonged,

for each boundary node i one defines the set Ni of the two duplicated boundarynodes in two-dimensional problems or of the Ni -folded boundary nodes in three-

dimensional problems.

Considering nowthesurfacial node-pairterm

1

2

i k

( fk fi ) in three-dimensionalproblems, a difficulty with the imposition of boundary conditions changing frominfow to outflow type across the common edge of two boundary triangles is en-

countered, similar to that just discussedfor thenodal metricvectors. Thisdifficulty

requires to split the two elemental contributions of each boundary metric vector

ik (see figure 4.3). Accordingly, elemental boundary metric vectors could be

defined as follows:

,eik =

ik e

i k n, i, k N , e E (i = k)

,e

i = i ei n, i N , e EIn terms of these elemental contributions the complete boundary term could be

written as follows:

BTi =

kNi,=

eEik

,eik

fe

k fe

i

2eEi

,ei f

e

i (4.18)

The double summation appearing in the first (node-pair) term is retained here to

have an expression valid in general for both two and three-dimensional problems.

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i

k

m

j

ne

nm

ei

mi

mj

ek

ik

i j

e

Figure 4.3: Boundary metric quantities for elements m and e in two spatial dimensions.

Thanks to the augmented set of boundary nodes just described, the elemental

contribution ,ei to the metric vector

i will be regarded as an independent nodal

metric vector and will be indicated by , to make explicit the fact that they belong

to the augmented set of boundary nodes. In this way, the term for the boundary

nodal contribution reduces to a summation over the artificially extended set of theduplicated (in 2D) or multiplicated (in 3D) boundary nodes, as follows:

eEi

,ei f

e

i Ni

f,

where are the boundary metric vectors duplicated (2D) or multiplicated (3D) in

conformity with the treatment of the boundary nodes. Note that with the assumed

notation the superscript e referring to the element becomes unnecessary.

4.4.2 Duplication of boundary edges in 3D problems

As anticipated, a similar treatment can be adopted also for the term associated

with the node-pair boundary contribution for three-dimensional equations. Each

boundary node-pair can be duplicated to account for the two boundary triangles

involved by the considered pair. The elemental contributions,e

i k are then consid-

ered as referring to an artificially doubled set of node-pairs on the boundary, that

will be indicated by N . (For the two dimensional equations N consists simplyof the set of all the node pairs on the boundary, whose number is equal to the

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number of edges along the boundary.) Thus, by interpreting the first summation

over k Ni,= as running over theduplicted set of boundary node-pairs, the secondsummation over the elemental contribution is made to disappear as follows:

kNi,=

eEik

,ei k

fe

k fe

i

2

k Ni,=

k

fk f2

where the number of the boundary metric vectors k

is twice that of edges on the

domain boundary in three dimensions. Again we note that the assumed notation of

the extended set of metric vectors

k

eliminate the need to refer to the elements in

the flux evaluation: in fact the number of flux values on the boundary is equal to thenumber of the augmented boundary nodes. The expression above is the general

form of the boundary contribution due to the surfacial node-pairs valid for the

three-dimensional equation. In the two-dimensionalcase there is no duplication of

the boundary edeges, which are parts of the discretized curve delimining the plane

computational domain. Therefore in two dimensions, the considered boundary

contribuition would reduce to the simpler standard summation

kNi,=

eEik

,eik

fe

k fe

i

2

kNi,=

i k fk fi

22D only.

In the following, the previous more general expression of this term will be always

considered to obtain equations suitable for implementing the numerical schemes

also in the three-dimensional case.

Both treatments for the boundary terms allow a very convenient simplification

of the program, only at the expense of interpreting the boundary as if it were

composed of fully disconnected edges in 2D or fully disconnected triangles in 3D.

The boundary contribution to the discrete equations will be written as follows

BTi= k Ni,=

k

f

k f2 Ni

f

.

It is worth reminding that in 2D the set Ni has always two elements while in 3Dit contains a number of elements equal to the number of triangles over the surface

which contain the boundary node i . On the other hand, the set N in 2D isnothing but N while in 3D it always contains twice the number of the boundary

edges of the mesh.

For subsequent reference, the discrete conservation law with the boundary

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terms expressed in the new form is written here explicitly:kNi

Mikduk

dt=

kNi,=

i k fi + fk

2

k Ni,=

k

fk f2

Ni

f.

(4.19)

Of course, the two terms associated with the boundary are present only when the

node i lies on the domain boundary: the absence of these terms for the discrete

equations associated to internal nodes is explicitly indicated by appending the

superscript to the boundary metric vectors k

and .It can be noted that the three summations in the right-hand side of the last

equation can be easily coded in the program by means of do-loops. The three

loops run on the three sets of all the node-pairs, all the (augmented) boundary

node-pairs and all the (augmented) boundary nodes, respectively. The superscript

e referring to the element is present in the two boundary terms simply due to the

fact that the flux on the boundary depends on the boundary value ofu, which in

turn depends on the inflow or outflow nature of each boundary element.

4.5 Diffusion term in node-pair formIn section 2.4, the finite element discretizationof a diffusion term has been derived;

in this section we address the problem of expressing the weak form of such a term

under the node-pair format. We anticipate that when the diffusion coefficient is

not constant, the resulting node-pair form can be achieved only at the expense of

introducing an additional error with respect to spatial discretization error of the

classical finite element procedure.

Let us now consider the weak form of the diffusion term. By integrating by

parts we have

i

i (uh ) = kNi

ik

(i ) k uk +

i

i b (4.20)

According to thestandard finite element methodthe twointegrals on the right-hand

side of this expression are computed by splitting them into a sum over the mesh

elements and a sum over the boundary sides, respectively. Let us examine how

these summations are evaluated by employing the same node-pair data structure

previously introduced.

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By exploiting the property kNe k(x) = 0, x e, on each finiteelement e of the mesh, we may calculate(i ) uh = (i )

kNe

ukk

=

kNe(i ) k uk

=

kNei,=(i ) k uk + (i ) i ui , x e.

By subtracting the term ui

i kNe k, which sum up to zero, we obtain

(i ) uh =

kNei,=(i ) k uk ui i

kNei,=

k

=

kNei,=(i ) k (uk ui ), x e

Therefore, the volume integral on the right-hand side of expression (4.20) for the

second-derivative can be rewritten as

i

(i ) uh = eEi

e

(i ) uh

=eEi

kNei,=

e

(i ) k

(uk ui )

=

kNi,=

ik

(i ) k

(uk ui )

where in the last relation we have used the fact that the integral vanishes outside

ik. Therefore, the Galerkin discretization of the diffusion term (4.20) isi

i (uh) =

kNi,=

i , k

ik

(uk ui ) +

i Ni b (4.21)

in which the node-pairs data structure has been put into evidence.

If we now approximate |ik with a constant value i k for each node pair i -k,i.e., for example, by taking

|ik ik = 12 (i + k), (4.22)

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we obtain the following approximationi

i (uh)

kNi,=iki ,k

ik

(uk ui ) +

i Ni b (4.23)

where the volume integral has been made independent from the function (x) and

thereforecan beevaluated once and forallat thebeginning of thecomputationas the

other metric vectors already introduced. Note that the considered approximation

preserves the symmetry of the stiffness matrix. Of course, for constant the

technique reduces to the exact finite element expression. Recalling the definition

(2.14) of the stiffness matrix element Ki k, the proposed approximation yieldsi

i (uh )

kNi,=ik Ki k (uk ui ) +

k(Ni NN)

ikN

i k

bk

(4.24)

where theboundary Neumann datumhas been replaced by its interpolant according

to the approximation

b(s, t)

k(Ni NN)bk(t)

k(s), s N. (4.25)

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Chapter 5

Finite-Volume method

on nonstructured meshes

In this chapter we study the finite volume method for unstructured meshes. We

derive theupwindscheme based on this spatial discretization technique. Thesemi-

discrete equations are first expressed in the standard finite-volume format. Then,

thy are expressed in the node-pair format, using the data structure and metric quan-

tities already described in Chapter 4, by taking advantage of Selmins theorem of

equivalence between finite volumes and finite elements in node-pair representation

[23].

5.1 Finite-Volume spatial discretization

Let us write the standard finite volume scheme for the scalar conservation law

(1.1). As well know, the finite volume method moves from the conservation law

written in the following integral form

d

dt

C

u(x, t) =

C

n f(u), C ,

where n indicates the outward normal vector of the region C . Let us nowconsider the discrete counterpart of the above equation by considering a certain

number offinite volumes Ci , with boundary Ci , each of them surrounding a single

node i of the triangulation of (see figure 5.1), namely,

d

dt

Ci

u(x, t) =

Ci

n f(u), i N. (5.1)

In the following, we assume that the finite volumes Ci satisfy the constraints:

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i Ci

Figure 5.1: Finite volume discretization of in two spatial dimensions. The underlyingFEM discretization using triangular meshes is shown (dashed lines).

Ci Ck = , i, k N, i = k, (5.2a)

kN

Ck = , i, k N, i = k, (5.2b)

i Ci i / Ck, i, k N, i = k. (5.2c)Condition (5.2a) assures that the open sets Ci , which are the internal parts of the

finite volumes Ci , are nonoverlapping, while condition (5.2c) implies that each

finite volume Ci is associated with a single node i . Alternatively, finite volumes

can be chosen to be coincident with the elements of the triangulation, that is,

Ce = e; the resulting scheme is said to be a cell-centred finite volume scheme,to be contrasted with the above node- or vertex-centred approach.

Over each finite volume Ci , the unknown u(x, t) is now approximated by itsspatial average as

u(x, t) ui (t) def= 1|Ci |Ci

u(x, t), x Ci . (5.3)

In other words, we consider the following piecewise constant approximation of

the unknown over (cf. figure 5.2), i.e.,

u(x, t) uh(x, t) def=kN

uk(t) Ik(x), (5.4)

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u

uh

Figure 5.2: Piecewise constant approximation uh ofu in one spatial dimension.

where, borrowing the nomenclature of finite elements, the shape functions are

chosen to be the characteristic functions Ik(x)

Ik(x) =

1 x Ck0 x / Ck

Therefore, if the finite volume does not change in time, we can write

d

dt

Ci

u(x, t) |Ci | duidt

.

As a first step toward the node-pair representation, the right hand side of (5.1) is

rearranged so as to put into evidence the node-pair structure of the data, namely,Ci

n f(u) =

kNi,=

Cik

ni f(u) +

Ci

n f(u), (5.5)

where ni denotes the outward normal with respect to the volume Ci . We notice

that ni = nk over Cik = Ci Ck. In the finite volume jargon, the set Cik isoften referred to as the cell interface between the volumes Ci and Ck (figure 5.3).

5.2 Upwind Finite-Volume scheme

We can now consider the issue of imposing inlet boundary conditions for the

hyperbolic problem. By means of the same procedure detailed in section 2.2, the

boundary condition is enforced in a weak sense throught the boundary integral by

substituting the unknown u with the function u defined in equation (2.7), which is

repeated here for convenience,

u(s, t)def=

u(s, t) if n(s) a(u(s, t)) 0a(s, t) if n(s) a(u(s, t)) < 0

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i kCi

CkCi k

ik

Figure 5.3: Cell interface Cik between the finite volumes Ci and Ck.

where a(u) = d f(u)/du and a(s, t) denotes the prescribed boundary value on theinlet portion in of the boundary. It should be reminded that a is the boundary

value prescribed on u and is not the intensity of the advection velocity a. This

boundary function is replaced by its piecewise constant approximation, as follows,

a(s, t) ah (s, t) =kN

ak(t) Ik (x),

where

Ik (s)def= Ik(x(s)), x(s) ,

denotes the trace of the shape function Ik over . The integral over the domainboundary in (5.5) is therefore evaluated as usual by the expression

Ci

n f =

Ci

n f(u). (5.6)

Due to the piecewise constant approximation considered, over Ci uh andah assume the constant values ui and ai , respectively, so that uh (s, t) is constant

over Ci as long as the outward normal n(s) is constant over Ci . Instandard finite volume discretizations, a constant (mean) normal is consideredover

Ci

, so that uh assumes a constant1 value and can therefore be factorized in

the computation of the boundary integral, namely,Ci

n f(uh(s)) fi (t)

Ci

n

with

fi (t)def= f(uh (s)) = constant s Ci .

1If a cell-centred volume is considered over a grid of linear elements, the normal n is indeed a

constant vector over Ce .

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Here, by taking advantage of the analysis performed in section 3.2, we choose to

consider a nonconstant normal over Ci and split the boundary integral asfollows

Ci

n f(uh(s)) =eE

i

fi (t)

Cie

n (5.7)

where e is the subregion of corresponding to element e of the finite element

triangulation and where we have introduced the following vector quantity

fi (t)def= f(u(si )) = constant s Ci e . (5.8)

Let us consider now the finite volume approximation of the domain integral.

First, we notice that the integrals Cik

ni f(u), (5.9)

are undefined at cell interfaces in the finite volume framework due to the piecewise

constant approximation uh chosen for u; with the discrete unknown uh discontin-

uous across Ci k. Therefore we introduce a numerical flux fik, representing an

approximation of f(u) at the cell interface Ci k, so as to have

Cikni f(u)

Cikni fi k.

If we select a constant fi k over Ci k, the numerical flux can be factorized in the

computation of the above integral to obtainCik

ni f(u) fi k

Cik

ni = fik ik,

where

ikdef=

Cik

ni .

We notice in passing that ik = ki . As an example, a second order approxima-tion of (5.9) can be obtained by choosing the following centred approximation off(u) over Cik

fi k =f(ui ) + f(uk)

2.

To allow for the inclusion of a more general approximation of (5.9), it is prefer-

able to introduce an integrated numerical function qi k, which depends on ui and

uk as well as on the integrated normal ik and which provides the following ap-

proximation Cik

ni f(u) qi k = q(ui , uk, i k).

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The integrated numerical flux qik is assumed to satisfy the following conditions:

q(ui , uk, i k) = q(uk, ui , ki ) (Conservation) (5.10a)q(u, u, i k) = f(u) i k (Consistency) (5.10b)

Condition (5.10a) is required to obtain a conservative scheme since it implies that

the integral of the flux computed at the cell interface Ci k is equal (up to the sign)

to the one computed at the cell interface Ck Ci . In terms of the numerical fluxfi k, the condition is

qik fik ik = fki ki qki ,

and it can be satisfied by selecting a symmetric numerical flux, namely, fik = fki ,since ik = ki .

The fully discrete form of (1.1) under the finite volume approximation is there-

fore,

|Ci | duidt

=

kNi,=q(ui , uk, ik)

eEi

,ei fi (t) (5.11)

where

,ei

def=

Ci en,

the functional form of the integrated numerical flux still remaining to be selected.

Consider now the following definition for the numerical flux

q(ui , uk, ik)def= ik

fi + fk2

12|ai k| (uk ui ) (5.12)

where a linearizationprojectedalong thedirection ik hasbeenintroduced, namely,

ai kdef=

i k fk fiuk ui

if uk = ui ,

i k d f(u)

du ui ukif uk = ui .

(5.13)

The integrated numerical flux (5.12) leads to well know first order (in space and

time) finite volume upwind scheme for the conservation law, namely,

|Ci |dui

dt=

kNi,=

ik

fi + fk2

12|ai k| (uk ui )

Ni

f, (5.14)

where the duplication or multiplication of the boundary nodes in 2D and 3D,

respectively, has been assumed to obtain the algorithmically more convenient rep-

resentation of the elemental contributions ,e

i to the averaged normal vectors.

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5.3 The bridge between finite volumes

and finite elements

Let us assume that a finite volume discretization of a two-dimensional domain is

derived from the underlying grid of triangular elements in the following way:

Each element e of the grid is divided into three subelements delimited by

the medians of the triangle itself, as in figure 5.4. The sets i , k and jassociated with thevertices i , kand j of triangle e, respectively, aredelimited

by the two element sides from each node and the two segments that join the

middle of each element side to the center of gravity xg of the element, where

xgdef= 1|e|

e

x.

The finite volume Ci associated with node i is given by the union of all the

subelements having i as a vertex, namely,

Cidef=

eEi

i .

An example of a finite volumeC

i assembled following the above prescriptions isgiven in figure 5.4.

In [23], the following identities are given for metric quantities generated over

the finite volume dicretization described above:

|Ci | =

i

i =

kNiMi k = L i , (5.15a)

ik = ik, (5.15b)i = i . (5.15c)

The proof of the important relation (5.15a) is direct. It can be obtained by con-sidering a representative triangle of the mesh and focusing on that part of its area

contributing to the cell associated to one its vertex, the node i in relation (5.15a).

This portion of the triangle is defined by the medians and its area and can be cal-

culated geometricaly. On the other hand, the contribution to the mass elements

Mik coming from the integrals inside the considered triangle can be evaluated

analytically for linear interpolations.

For instance, let us consider the representative generic triangle with vertices

x1 = (0, 0), x2 = (a, 0) and x3 = (b, c), and let us takethe part inside this triangle

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i

k

j

Ciei

e

k

ej

e

Figure 5.4: Construction of the finite volume Ci satisfying the finite elements/finite vol-umes equivalence requirements.

of cell around the node (0, 0) coincident with its first vertex. The intersection of

the medians of the triangle is easily find to be xm

=((a

+b)/3, c/3). Thus the

area A of the cell portion inside the triangle is made of two components and isgiven by

A = 12

a

2ym +

1

2

b2 + c2

2d

where d is the distance of xm to the triangle side oblique with respect to the

Cartesian axes. Since d = 13

ac/

b2 + c2, we obtain

A = ac12

+ ac12

= ac6

.

To evaluate (the contribution to) the mass elements M1,1, M1,2 and M1,3 the basis

functions over the considered triangle are needed. The three linear interpolation

functions are easily found to be

1(x, y) = 1 x

a

1 ba

yc

,

2(x, y) =x

a by

ac,

3(x, y) =y

c.

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The integrals contributing to the mass elements are triangle

1k dx dy =c

0

d y

a+(b1)y/cby /c

1(x, y) k(x , y) d x.

By a change of variables y Y = y/c and x X = (x b)/a the threeintegrals give the mass contributions

M1,1 =ac

12, M1,2 =

ac

24, M1,3 =

ac

24,

which add to geometrical result A = ac/6, as required. This proof can beextended in three dimensions to tetrahedral elemenst with linear interpolations.

As a consequence, the upwind finite volume scheme (5.14) can be written in

terms of the finite element metric quantities introduced so far as

Lidui

dt=

kNi,=

i k

fi + fk2

12|aik| (uk ui )

Ni

f, (5.16)

where we recall the definition of the linearization projected along the direction ik

aik def= ik

fk fiuk

ui

if uk = ui

ik d f(u)

du

ui uk

if uk = ui .

It can be noticed that the above upwind finite volume method can be written in the

usual conservation form, i.e.,

L idui

dt=

kNi,=

qi k Ni

f,

by introducing the integrated numerical flux

qi k = i k fi + fk

2 1

2|aik| (uk ui ).

This result is importantbecause it shows that theunstructuredfinitevolume upwind

method presented here extends to multidimensional equations the schemes written

in conservation form for one-dimensional problems.

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Chapter 6

Taylor-series-based integration

schemes

In this chapter we describe how to compute the terms occurring in LaxWendroff

scheme as well as in TaylorGalerkin scheme of Donea [3]. First, we examine

the second-order correction term arising from the Taylor expansion in time of the

multidimensional conservation law (1.1) for a scalar quantity u. Then, the Taylor

Galerkin scheme for a conservation law endowed with a linear flux function is

derived. In Appendix D, the numerical properties of the original TaylorGalerkin

scheme and of some two-step versions of it are recalled.

6.1 Nonlinear scalar conservation law

The TaylorGalerkin scheme is based on the idea of performing the discretization

in time before performing thediscretization in space; theformeris accomplished by

means of a Taylor expansion at time level n, while the latter is performed by means

of the standard Galerkin finite element method. More precisely, the unknown at

the new time level tn+1 is obtained by means of the following Taylor expansion attime tn

un+1 = un + ttun + 12 (t)2 2t un + 16 (t)3 3t un + O(t)4,where, with standard notation, the subscript indicates derivation with respect to

time. The terms involving the time derivative are obtained from the governing

equation, i.e.,

tu = f(u),

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and by taking the time derivatives of the same equation to give,

2t u = t f =

t f = d f

dutu

=

d f

du( f)

= (a f),

where we have substituted for the advection velocity a = a(u) = d f(u)/du. Theweak form of the term (a f) is therefore easily found, using integration by

parts, in the form

, (a f)

=

, a f

+

n a f

= a , a u+ n a f,or, indicating explicitly the dependence on the unknown u,

, [a(u) f(u)]= a(u) , a(u) u+ n a(u) f(u). (6.1)

Thus, for the nonlinear conservation law tu + f(u) = 0, the second-orderterm of the TaylorGalerkin method gives a symmetric correction. This term is

the weak variational expression of the second-order directional derivative, along

the (local) direction of the advection field a(u).To simplify the exposition, the description of TG schemes considered so far

has not examined how to impose the boundary condition. We now discuss this

point and we distinguishbetween the case of a nonlinear conservation law and that

of a conservation law with a linear flux.

For the nonlinear conservation law, there is only one boundary term, namely, n a(u) a(u) u,

stemming from the integration by parts. Note that in this expression the vectoru

most be evaluated on the boundary where, in general, has a nonzero normal

component. The determination of this component involves also values ofu atinternal points of.

The semi-discrete equations for the TaylorGalerkin method for the scalar

conservation law reads

(, un+1 un)t

= , f(un)

n f(un)

t2

an, anun

+ t2

n an anun

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6.2 The fully discrete form of the equations

Let us consider first the second-order TaylorGalerkin scheme (TG2 or L

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