Lecture 1: Finite Volume WENO Schemes

Lecture 1: Finite Volume WENO Schemes

Chi-Wang Shu

Division of Applied Mathematics

Brown University

FINITE VOLUME WENO SCHEMES

Outline of the First Lecture

• General description of finite volume schemes for conservation laws

• The WENO reconstruction procedure

• Bound-preserving limiter for high order finite volume WENO schemes

• A simple WENO limiter for discontinuous Galerkin methods

• Concluding remarks

Division of Applied Mathematics, Brown University


Finite volume schemes for conservation laws

We look first at the one-dimensional hyperbolic conservation law

ut + f(u)x = 0

which has discontinuous solutions even if the initial condition is smooth.

We discretize the computational domain into cells Ii = [xi−1/2, xi+1/2]

with cell sizes ∆xi (not necessary to be uniform or smooth varying). The

cell averages are denoted by

ui =1

∆xi

∫ xi+1/2

xi−1/2

u(x)dx.



A finite volume scheme approximates this conservation law in its integral

formd

dtui +

1

∆xi

(

f(ui+1/2) − f(ui−1/2))

= 0 (1)

To convert (1) to a finite volume scheme, we take our computational

variables as the cell averages

ui, i = 1, 2, · · · , N

and use a reconstruction procedure to obtain an approximation to ui+1/2.



A typical reconstruction procedure is to choose several consecutive cells

near xi+1/2, which typically include at least one of Ii and Ii+1, the two

neighbors of xi+1/2. The collection of these cells are called the stencil of

the reconstruction. We seek a polynomial p(x) (or another simple function

such as a trigonometric or exponential function) whose cell average over

each cell Ij in the stencil agrees with the given cell average uj . We then

take ui+1/2 = p(xi+1/2).



In order to obey upwinding for stability, we replace f(ui+1/2) by

f(

u−

i+1/2, u+i+1/2

)

where f(u−, u+) is a monotone numerical flux satisfying

1. f(u−, u+) is non-decreasing in its first argument u− and

non-increasing in its second argument u+, symbolically f(↑, ↓);

2. f(u−, u+) is consistent with the physical flux f(u), i.e.

f(u, u) = f(u);

3. f(u−, u+) is Lipschitz continuous with respect to both arguments u−

and u+.

Here, both u−

i+1/2 and u+i+1/2 are obtained through the reconstruction

procedure, with their stencil biased to the left and to the right, respectively.



Time discretization can be achieved by the TVD (also called SSP)

Runge-Kutta or multi-step methods Shu and Osher JCP 1987; Gottlieb,

Shu and Tadmor SIAM Rev 2001; Gottlieb, Ketcheson and Shu, World

Scientific 2011. For example, the third order TVD Runge-Kutta scheme is

u(1) = un + ∆tL(un)

u(2) =3

4un +

1

4u(1) +

1

4∆tL(u(1))

un+1 =1

3un +

2

3u(2) +

2

3∆tL(u(2))



One advantage of finite volume schemes is that they can be generalized to

multi-dimensions including unstructured meshes “easily” in principle, even

though the reconstruction procedure and the computation of numerical

fluxes become more complicated, especially for unstructured meshes.



The WENO reconstruction procedure

We would like to have schemes which are both

• high order accurate in smooth regions, and

• (essentially) non-oscillatory with sharp shock transition

For typical linear schemes, i.e. schemes which are linear for a linear PDE

ut + aux = 0 (2)

(this corresponds to the situation that the stencil relative to the point xi+1/2

is fixed, for example, it is always {Ii−1, Ii, Ii+1}), the two properties

desired above cannot be fulfilled simultaneously (Godunov Theorem).



We would therefore need to consider nonlinear schemes, which are

nonlinear even for the linear PDE (2), such as the WENO schemes. In

fact, the nonlinearity of the algorithm is only at the stage of choosing

stencils in the reconstruction.

Essentially non-oscillatory (ENO) reconstruction (Harten, Engquist, Osher

and Chakravarthy, JCP 1987):

• Uniform high order polynomial reconstruction;

• The stencil is locally adaptive: among several candidate stencils one

is chosen according to local smoothness.



j-2

j-1

j

j+1

j+2

S0

S1

S2

j+1/2

Figure 1: Three possible stencils for reconstructing the point value at

xj+1/2 using three cells in each stencil.



Weighted ENO (WENO) reconstruction (Liu, Osher and Chan, JCP 1994;

Jiang and Shu, JCP 1996):

• Instead of using just one candidate stencil; a linear combination of all

candidate stencils is used

• The choice of the weight to each candidate stencil, which is a

nonlinear function of the cell averages, is a key to the success of

WENO.



Advantages of ENO and WENO schemes:

• Uniform high order accuracy in smooth regions including at smooth

extrema, unlike second order TVD schemes which degenerate to first

order accuracy at smooth extrema;

• Sharp and essentially non-oscillatory (to the eyes) shock transition;

• Robust for many physical systems with strong shocks;

• Especially suitable for simulating solutions containing both

discontinuities and complicated smooth solution structure, such as

shock interaction with vortices.



Some advantages of WENO schemes over ENO schemes:

• Higher order of accuracy with the same set of candidate stencils: the

order of accuracy is 3 instead of 2 for piecewise linear, and 5 instead

of 3 for piecewise quadratic;

• No logical “if” statements in the stencil choosing process of ENO.

Cleaner programming;

• Numerical flux function is smoother: C∞ instead of only Lipschitz as

in the ENO case. Hence: (i) a convergence proof when the solution is

smooth (Jiang and Shu, JCP 1996), and (ii) better steady state

convergence (Zhang and Shu, JSC 2007; Zhang, Jiang and Shu, JSC

2011; Hao et al., JCP submitted).



The WENO reconstruction procedure

Given the cell averages ui = 1∆xi

∫ xi+1/2

xi−1/2u(x)dx of a piecewise smooth

function u(x) for the cells Ii = [xi−1/2, xi+1/2] with cell sizes ∆xi, find

an approximation to the function u(x) at a desired location, e.g. at the cell

boundaries xi+1/2.

General procedure of reconstruction with a given stencil. e.g. to

reconstruct ui+1/2 given ui−1, ui and ui+1:



1. Find the unique second order polynomial p(x) which agrees with the

three given cell averages ui−1, ui and ui+1 for the three cells in the

stencil, respectively:

1

∆xi−1

∫ xi−1/2

xi−3/2

p(x)dx = ui−1,

1

∆xi

∫ xi+1/2

xi−1/2

p(x)dx = ui,

1

∆xi+1

∫ xi+3/2

xi+1/2

p(x)dx = ui+1.

2. Take the value p(xi+1/2) as an approximation to ui+1/2:

ui+1/2 = p(xi+1/2)



3. The approximation ui+1/2 can be written out eventually as a linear

combination of the given cell averages ui−1, ui and ui+1 because the

procedure is linear:

ui+1/2 = −1

6ui−1 +

5

6ui +

1

3ui+1

This approximation is third order accurate if the function u(x) is smooth in

the stencil {Ii−1, Ii, Ii+1}.



Using such approximations with a fixed stencil leads to high order linear

schemes, which will be oscillatory in the presence of shocks by the

Godunov Theorem.

The general procedure of a WENO reconstruction:

1. Compute the approximations from several different substencils, e.g.

the three stencils in Figure 2:



j-2

j-1

j

j+1

j+2

S0

S1

S2

j+1/2

Figure 2: Three sub-stencils for the reconstruction at xj+1/2 using three

cells in each stencil.



u(0)i+1/2 =

1

3ui−2 −

7

6ui−1 +

11

6ui

u(1)i+1/2 = −

1

6ui−1 +

5

6ui +

1

3ui+1

u(2)i+1/2 =

1

3ui +

5

6ui+1 −

1

6ui+2

If the function u(x) is smooth in all three substencils, then the three

approximations u(0)i+1/2, u

(1)i+1/2 and u

(2)i+1/2 are all third order accurate.



2. Find the combination coefficients γ0, γ1 and γ2, also called linear

weights, such that the linear combination

ui+1/2 = γ0u(0)i+1/2 + γ1u

(1)i+1/2 + γ2u

(2)i+1/2

is fifth order accurate if u(x) is smooth in all substencils. This can be

easily achieved with γ0 = 110

, γ1 = 35

and γ2 = 310

:

ui+1/2 =1

10u

(0)i+1/2 +

3

5u

(1)i+1/2 +

3

10u

(2)i+1/2

would lead to a fifth order accurate linear scheme which is oscillatory.

At this stage, if we only require the reconstructed value ui+1/2 to be of

the same order of accuracy as that from each of the substencils (in

this case, third order rather than fifth order), then we can choose the

linear weights γk > 0 arbitrarily as long as they sum to one.



3. Find the nonlinear weights w0, w1 and w2 such that

ui+1/2 = w0u(0)i+1/2 + w1u

(1)i+1/2 + w2u

(2)i+1/2

is both fifth order accurate in smooth regions and non-oscillatory for

shocks. Thus we require the nonlinear weights w0, w1 and w2 to

satisfy the following two properties:

• If u(x) is smooth in all three substencils, then the nonlinear

weights w0, w1 and w2 are close to the linear weights γ0, γ1 and

γ2:

wk = γk + O(∆x2), k = 0, 1, 2.

• If u(x) has a discontinuity in the substencil Sk, then the

corresponding wk is very small:

wk = O(∆x4)



4. A robust choice of the nonlinear weights, given in (Jiang and Shu, JCP

1996) and used in most WENO literature, is

wk =wk

w0 + w1 + w2, wk =

γk

(ε + βk)2

where ε = 10−6 typically (it can be adjusted by the average size of

the solution), and the smoothness indicator βk measures the

smoothness of the function u(x) in the substencil Sk and is given by

βk = ∆xi

∫ xi+1

2

xi− 1

2

(p′r(x))2dx + ∆x3i

∫ xi+1

2

xi− 1

2

(p′′r(x))2dx.



These smoothness indicators can be worked out explicitly as

β0 =13

12(ui−2 − 2ui−1 + ui)

2 +1

4(ui−2 − 4ui−1 + 3ui)

2

β1 =13

12(ui−1 − 2ui + ui+1)

2 +1

4(ui−1 − ui+1)

2

β2 =13

12(ui − 2ui+1 + ui+2)

2 +1

4(3ui − 4ui+1 + ui+2)

2



Bound-preserving limiter

For many physical problems, there are natural bounds for the solution. For

example, if the solution represents a percentage of a component in a

mixture, then it must be between 0 and 1. If the solution is the probability

density function, it must be non-negative. For Euler equations of gas

dynamics, the density and pressure must be non-negative. For shallow

water equations, the water height should be non-negative, etc.



We take the scalar conservation laws

ut + ▽ · F(u) = 0, u(x, 0) = u0(x) (3)

as an example. An important property of the entropy solution (which may

be discontinuous) is that it satisfies a strict maximum principle: If

M = maxx

u0(x), m = minx

u0(x), (4)

then u(x, t) ∈ [m,M ] for any x and t.



First order monotone schemes can maintain the maximum principle. For

the one-dimensional conservation law

ut + f(u)x = 0,

the first order monotone scheme

un+1j = Hλ(u

nj−1, u

nj , un

j+1)

= unj − λ[h(un

j , unj+1) − h(un

j−1, unj )]

where λ = ∆t∆x

and h(u−, u+) is a monotone flux (h(↑, ↓)), satisfies

Hλ(↑, ↑, ↑)

under a suitable CFL condition

λ ≤ λ0.



Therefore, if

m ≤ unj−1, u

nj , un

j+1 ≤ M

then

un+1j = Hλ(u

nj−1, u

nj , un

j+1) ≥ Hλ(m,m,m) = m,

and

un+1j = Hλ(u

nj−1, u

nj , un

j+1) ≤ Hλ(M,M,M) = M.



However, for higher order linear schemes, i.e. schemes which are linear

for a linear PDE

ut + aux = 0

for example the second order accurate Lax-Wendroff scheme

un+1j =

aλ

2(1 + aλ)un

j−1 + (1 − a2λ2)unj −

aλ

2(1 − aλ)un

j+1

where λ = ∆t∆x

and |a|λ ≤ 1, the maximum principle is not satisfied. In

fact, no linear schemes with order of accuracy higher than one can satisfy

the maximum principle (Godunov Theorem).



Therefore, nonlinear schemes, namely schemes which are nonlinear even

for linear PDEs, have been designed to overcome this difficulty. These

include roughly two classes of schemes:

• TVD schemes. Most TVD (total variation diminishing) schemes also

satisfy strict maximum principle, even in multi-dimensions. TVD

schemes can be designed for any formal order of accuracy for

solutions in smooth, monotone regions. However, all TVD schemes

will degenerate to first order accuracy at smooth extrema.

• TVB schemes, ENO schemes, WENO schemes. These schemes do

not insist on strict TVD properties, therefore they do not satisfy strict

maximum principles, although they can be designed to be arbitrarily

high order accurate for smooth solutions.



Remark: If we insist on the maximum principle interpreted as

m ≤ un+1j ≤ M, ∀j

if

m ≤ unj ≤ M, ∀j,

where unj is either the approximation to the point value u(xj, t

n) for a

finite difference scheme, or to the cell average 1∆x

∫ xj+1/2

xj−1/2u(x, tn)dx for

a finite volume or DG scheme, then the scheme can be at most second

order accurate (proof due to Harten, see Zhang and Shu, Proceedings of

the Royal Society A, 2011).



Therefore, the correct procedure to follow in designing high order schemes

that satisfy a strict maximum principle is to change the definition of

maximum principle. Note that a high order finite volume scheme has the

following algorithm flowchart:

(1) Given {unj }

(2) reconstruct un(x) (piecewise polynomial with cell average unj )

(3) evolve by, e.g. Runge-Kutta time discretization to get {un+1j }

(4) return to (1)



Therefore, instead of requiring

m ≤ un+1j ≤ M, ∀j

if

m ≤ unj ≤ M, ∀j,

we will require

m ≤ un+1(x) ≤ M, ∀x

if

m ≤ un(x) ≤ M, ∀x.

Similar definition and procedure can be used for discontinuous Galerkin

schemes.



Maximum-principle-preserving for scalar equations

The flowchart for designing a high order scheme which obeys a strict

maximum principle is as follows:

1. Start with un(x) which is high order accurate

|u(x, tn) − un(x)| ≤ C∆xp

and satisfy

m ≤ un(x) ≤ M, ∀x

therefore of course we also have

m ≤ unj ≤ M, ∀j.



2. Evolve for one time step to get

m ≤ un+1j ≤ M, ∀j. (5)

3. Given (5) above, obtain the reconstruction un+1(x) which

• satisfies the maximum principle

m ≤ un+1(x) ≤ M, ∀x;

• is high order accurate

|u(x, tn+1) − un+1(x)| ≤ C∆xp.



Three major difficulties

1. The first difficulty is how to evolve in time for one time step to

guarantee

m ≤ un+1j ≤ M, ∀j. (6)

This is very difficult to achieve. Previous works use one of the

following two approaches:



• Use exact time evolution. This can guarantee

m ≤ un+1j ≤ M, ∀j.

However, it can only be implemented with reasonable cost for linear

PDEs, or for nonlinear PDEs in one dimension. This approach was

used in, e.g., Jiang and Tadmor, SISC 1998; Liu and Osher,

SINUM 1996; Sanders, Math Comp 1988; Qiu and Shu, SINUM

2008; Zhang and Shu, SINUM 2010; to obtain TVD schemes or

maximum-principle-preserving schemes for linear and nonlinear

PDEs in one dimension or for linear PDEs in multi-dimensions, for

second or third order accurate schemes.



• Use simple time evolution such as SSP Runge-Kutta or multi-step

methods. However, additional limiting will be needed on un(x)

which will destroy accuracy near smooth extrema.

We have figured out a way to obtain

m ≤ un+1j ≤ M, ∀j

with simple Euler forward or SSP Runge-Kutta or multi-step methods

without losing accuracy on the limited un(x):



The evolution of the cell average for a higher order finite volume or DG

scheme satisfies

un+1j = G(un

j , u−

j− 1

2

, u+j− 1

2

, u−

j+ 1

2

,u+j+ 1

2

)

= unj − λ[h(u−

j+ 1

2

, u+j+ 1

2

) − h(u−

j− 1

2

, u+j− 1

2

)],

where

G(↑, ↑, ↓, ↓, ↑)

therefore there is no maximum principle. The problem is with the two

arguments u+j− 1

2

and u−

j+ 1

2

which are values at points inside the cell

Ij .



The polynomial pj(x) (either reconstructed in a finite volume method

or evolved in a DG method) is of degree k, defined on Ij such that unj

is its cell average on Ij , u+j− 1

2

= pj(xj− 1

2

) and u−

j+ 1

2

= pj(xj+ 1

2

).

We take a Legendre Gauss-Lobatto quadrature rule which is exact for

polynomials of degree k, then

unj =

m∑

ℓ=0

ωℓpj(yℓ)

with y0 = xj− 1

2

, ym = xj+ 1

2

. The scheme for the cell average is then

rewritten as



un+1j = ωm

[

u−

j+ 1

2

−λ

ωm

(

h(u−

j+ 1

2

, u+j+ 1

2

) − h(u+j− 1

2

, u−

j+ 1

2

))

]

+ω0

[

u+j− 1

2

−λ

ω0

(

h(u+j− 1

2

, u−

j+ 1

2

) − h(u−

j− 1

2

, u+j− 1

2

))

]

+

m−1∑

ℓ=1

ωℓpj(yℓ)

= Hλ/ωm(u+j− 1

2

, u−

j+ 1

2

, u+j+ 1

2

) + Hλ/ω0(u−

j− 1

2

, u+j− 1

2

, u−

j+ 1

2

)

+m−1∑

ℓ=1

ωℓpj(yℓ).



Therefore, if

m ≤ pj(yℓ) ≤ M

at all Legendre Gauss-Lobatto quadrature points and a reduced CFL

condition

λ/ωm = λ/ω0 ≤ λ0

is satisfied, then

m ≤ un+1j ≤ M.



2. The second difficulty is: given

m ≤ un+1j ≤ M, ∀j

how to obtain an accurate reconstruction un+1(x) which satisfy

m ≤ un+1(x) ≤ M, ∀x.

Previous work was mainly for relatively lower order schemes (second

or third order accurate), and would typically require an evaluation of

the extrema of un+1(x), which, for a piecewise polynomial of higher

degree, is quite costly.

We have figured out a way to obtain such reconstruction with a very

simple scaling limiter, which only requires the evaluation of un+1(x)

at certain pre-determined quadrature points and does not destroy

accuracy:



We replace pj(x) by the limited polynomial pj(x) defined by

pj(x) = θj(pj(x) − unj ) + un

j

where

θj = min

{∣

∣

∣

∣

M − unj

Mj − unj

∣

∣

∣

∣

,

∣

∣

∣

∣

m − unj

mj − unj

∣

∣

∣

∣

, 1

}

,

with

Mj = maxx∈Sj

pj(x), mj = minx∈Sj

pj(x)

where Sj is the set of Legendre Gauss-Lobatto quadrature points of

cell Ij .

Clearly, this limiter is just a simple scaling of the original polynomial

around its average.



The following lemma, guaranteeing the maintenance of accuracy of

this simple limiter, is proved in Zhang and Shu, JCP 2010a:

Lemma: Assume unj ∈ [m,M ] and pj(x) is an O(∆xp)

approximation, then pj(x) is also an O(∆xp) approximation.



3. The third difficulty is how to generalize the algorithm and result to 2D

(or higher dimensions). Algorithms which would require an evaluation

of the extrema of the reconstructed polynomials un+1(x, y) would not

be easy to generalize at all.

Our algorithm uses only explicit Euler forward or SSP (also called

TVD) Runge-Kutta or multi-step time discretizations, and a simple

scaling limiter involving just evaluation of the polynomial at certain

quadrature points, hence easily generalizes to 2D or higher

dimensions on structured or unstructured meshes, with strict

maximum-principle-satisfying property and provable high order

accuracy.



The technique has been generalized to the following situations maintaining

uniformly high order accuracy:

• 2D scalar conservation laws on rectangular or triangular meshes with

strict maximum principle (Zhang and Shu, JCP 2010a; Zhang, Xia and

Shu, JSC 2012).

• 2D incompressible equations in the vorticity-streamfunction

formulation (with strict maximum principle for the vorticity), and 2D

passive convections in a divergence-free velocity field, i.e.

ωt + (uω)x + (vω)x = 0,

with a given divergence-free velocity field (u, v), again with strict

maximum principle (Zhang and Shu, JCP 2010a; Zhang, Xia and Shu,

JSC 2012).



The framework of establishing maximum-principle-satisfying schemes for

scalar equations can be generalized to hyperbolic systems to preserve the

positivity of certain physical quantities, such as density and pressure of

compressible gas dynamics.

Positivity-preserving finite volume or DG schemes have been designed for:

• One and multi-dimensional compressible Euler equations maintaining

positivity of density and pressure (Zhang and Shu, JCP 2010b; Zhang,

Xia and Shu, JSC 2012).

• One and two-dimensional shallow water equations maintaining

non-negativity of water height and well-balancedness for problems

with dry areas (Xing, Zhang and Shu, Advances in Water Resources

2010; Xing and Shu, Advances in Water Resources 2011).



• One and multi-dimensional compressible Euler equations with source

terms (geometric, gravity, chemical reaction, radiative cooling)

maintaining positivity of density and pressure (Zhang and Shu, JCP

2011).

• One and multi-dimensional compressible Euler equations with

gaseous detonations maintaining positivity of density, pressure and

reactant mass fraction, with a new and simplified implementation of

the pressure limiter. DG computations are stable without using the

TVB limiter (Wang, Zhang, Shu and Ning, JCP 2012).

• A minimum entropy principle satisfying high order scheme for gas

dynamics equations (Zhang and Shu, Num Math 2012).



A simple WENO limiter for DG methods

Finite volume WENO schemes involve rather complicated reconstruction

procedure for unstructured meshes. However, they have the advantage of

essentially non-oscillatory performance for solutions with strong shocks.

Therefore, the WENO methodology is often used as limiters for

discontinuous Galerkin (DG) methods (Qiu and Shu, JCP 2003; SISC

2005; Computers & Fluids 2005; Zhu, Qiu, Shu and Dumbser, JCP 2008;

Zhu and Qiu, JCP 2012.).



In particular, the very recent work in Zhong and Shu, JCP 2012; Zhu,

Zhong, Shu and Qiu, JCP submitted contains a very simple and effective

WENO limiter:

• Use a troubled-cell indicator to identify troubled cells. Qiu and Shu,

SISC 2005.

• If the cell Ij is identified as a troubled cell, then the DG solution

polynomial pj(x) is replaced by a convex combination of pj(x) with

pj−1(x) and pj+1(x), the DG solution polynomials of the two

immediate neighboring cells. Suitable adjustment is made (a constant

is added to pj−1(x) to obtain pj−1(x), likewise for pj+1(x)) to

ensure that the new polynomial maintains the original cell average

(conservation).



• Details:

pnewj = w1pj−1(x) + w2pj(x) + w3pj+1(x)

where

wℓ =wℓ

w1 + w2 + w3; wℓ =

γℓ

(sℓ + ε)2

with the linear weights given by

γ1 = γ3 =1

1000, γ2 =

998

1000

and the sℓ are the standard smoothness indicators of WENO

approximations.



Example 1: A Mach 3 wind tunnel with a step. The wind tunnel is 1 length

unit wide and 3 length units long. The step is 0.2 length units high and is

located 0.6 length units from the left-hand end of the tunnel. The problem

is initialized by a right-going Mach 3 flow. Reflective boundary conditions

are applied along the wall of the tunnel and inflow/outflow boundary

conditions are applied at the entrance/exit.



0 1 2 3X

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

Figure 3: Forward step problem. Sample mesh. The mesh points on the

boundary are uniformly distributed with cell length h = 1/20.



0 1 2 3X

0

0.5

1

Y

Figure 4: Forward step problem. Third order (k = 2) RKDG with the

WENO limiter. 30 equally spaced density contours from 0.32 to 6.15. The

mesh points on the boundary are uniformly distributed with cell length h =

1/100.



0 1 2 3X

0

0.5

1

Y

Figure 5: Forward step problem. Third order (k = 2) RKDG with the

WENO limiter. Troubled cells. Circles denote triangles which are identified

as troubled cell subject to the WENO limiting. The mesh points on the

boundary are uniformly distributed with cell length h = 1/100.



Example 2: We consider inviscid Euler transonic flow past a single

NACA0012 airfoil configuration with Mach number M∞ = 0.85, angle of

attack α = 1◦. The computational domain is [−15, 15] × [−15, 15].



-1 0 1 2X/C

-1

-0.5

0

0.5

1

1.5

Y/C

Figure 6: NACA0012 airfoil mesh zoom in.



-1 0 1 2X/C

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Y/C

-1 0 1 2X/C

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Y/C

Figure 7: NACA0012 airfoil. Mach number. M∞ = 0.85, angle of attack

α = 1◦, 30 equally spaced mach number contours from 0.158 to 1.357.

Left: second order (k = 1); right: third order (k = 2) RKDG with the

WENO limiter.



-1 -0.5 0 0.5 1 1.5 2X/C

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Y

/C

-1 -0.5 0 0.5 1 1.5 2X/C

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Y/C

Figure 8: NACA0012 airfoil. Troubled cells. Circles denote triangles which

are identified as troubled cells subject to the WENO limiting. M∞ = 0.85,

angle of attack α = 1◦. Left: second order (k = 1); right: third order

(k = 2) RKDG with the WENO limiter.



Concluding remarks

• Finite volume schemes can maintain conservation and achieve high

order accuracy both for structured and unstructured meshes.

• WENO reconstruction in finite volume schemes can provide high order

accuracy and essentially non-oscillatory shock transition.

• Bound-preserving limiter based on a simple scaling limiter can

guarantee maximum-principle for scalar equations and

passive-convection in a divergence-free velocity field, positivity for

density and pressure for Euler equations, and positivity for water

height for shallow water equations, among many others applications,

without compromising high order accuracy.



• A simple WENO limiter can be designed for discontinuous Galerkin

method to handle strong shocks without affecting high order accuracy.



The End

THANK YOU!


Documents

Lecture 1: Finite Volume WENO Schemes