Upwind and central WENO schemes

Applied Mathematics and Computation 166 (2005) 434–448

www.elsevier.com/locate/amc

Upwind and central WENO schemes

Lei Tang

ZONA Technology, Inc., 7430 E. Stetson Drive, Suite 205, Scottsdale, AZ 85251, USA

Abstract

There are two different ways to construct a WENO (weighted ENO) scheme for

numerical solution of hyperbolic conservation laws. The more popular approach is to

construct a WENO interpolation directly for computing the interface values of the solu-

tion. The resulting hyperbolic solver is upstream central in the smooth regions when the

nonlinear weights reduce to the optimal linear weights. We refer it to as upwind WENO

scheme. On the other hand, based on the same set of multiple lower-order polynomials,

the so-called ‘‘Reconstruction via a Primitive Function’’ approach constructs a WENO

interpolation to first approximate the derivative of the primitive function of the solu-

tion. The resulting WENO interpolation of the interface values of the solution is sym-

metric with respect to the interface and leads to a central hyperbolic solver in the

smooth regions when the nonlinear weights reduce to the optimal linear weights. We

refer it to as central WENO scheme. However, both types of WENO schemes are

upwind in the non-smooth regions and therefore stable. The stability of a central

WENO scheme implies that upwinding is only needed in the non-smooth regions.

� 2004 Elsevier Inc. All rights reserved.

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doi:10.1016/j.amc.2004.06.043

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L. Tang / Appl. Math. Comput. 166 (2005) 434–448 435

1. Introduction

During the last three decades, a large amount of efforts have been spent in

the computational society to remove the numerical oscillations created by a

high-order numerical discretization of the hyperbolic conservation laws near

a discontinuity. Originally people suppress numerical oscillations created by

central schemes through the introduction of an artificial viscosity (e.g.,

[1,2]). However, fine-tuning the size of this artificial viscosity is problem-dependent. Later, advanced upwind schemes become more popular. The so-

called FCT (flux corrected transport) and TVD (total variation diminishing)

schemes (e.g., [3–10]) are the first high-order upwind schemes, which achieve

monotonicity preservation. To diminish the total variation of the solution,

however, these schemes degrade to first-order accuracy at local extrema

although they can be highly accurate elsewhere. Thus the so-called ENO

(essentially non-oscillatory) schemes (e.g., [11–13]) have been further devel-

oped to achieve uniformly high-order accuracy. By using a linear combina-tion of all candidate stencils instead of only one as in the ENO schemes,

the more recently developed WENO (weighted ENO) schemes (e.g., [14–

19]) further improve the accuracy and convergence property over the ENO

schemes.

The WENO schemes are first developed by Liu et al. [14] for the cell-aver-

aged-value approach. Later, Jiang and Shu further improve the schemes and

extend them to the nodal-value approach in [15]. Especially a general frame-

work is given in [15] to design the smoothness indicators, which is the keyfor construction of a WENO scheme. Higher-order extensions of these schemes

have been shown in [16]. An important feature of these WENO schemes is that

when the nonlinear weights reduce to the optimal linear weights in the smooth

regions, they become upstream central. We refer this type of WENO schemes

to as upwind WENO schemes.

Following the works of Davis [20] and Jiang et al. [17], in this paper, we will

develop another type of WENO interpolation based on the same set of multiple

lower-order polynomials as an upwind WENO scheme using the so-called‘‘Reconstruction via a Primitive Function’’ technique, which is originally used

for construction of an ENO scheme [11]. As a result, we yield a central hyper-

bolic solver in the smooth regions when the nonlinear weights reduce to the

optimal linear weights, referred to as a central WENO scheme. Different from

those central WENO schemes using staggered meshes in [18,19], which still

have some upwinding flavors in the smooth regions through the evolution

stage, the central WENO scheme developed here is completely free from the

evolution stage and thereby central in the smooth regions although upwindin the non-smooth regions. Based on the local smoothness measures of the

solution, this type of WENO schemes achieves a real transition between central

and upwind schemes.

436 L. Tang / Appl. Math. Comput. 166 (2005) 434–448

We will start with a brief introduction of the Godunov-type schemes in Sec-

tion 2 and the well-known upwind WENO5 scheme in Section 3. Then, based

on the same set of multiple lower-order polynomials as the upwind WENO5

scheme, a central WENO4 scheme will be further constructed in Section 4

and compared with the upwind WENO5 scheme. Finally, Section 5 will present

several typical test cases to compare the accuracy of these two types of WENO

schemes.

2. Basic formulation

Given the one-dimensional scalar conservation law

ut þ f ðuÞx ¼ 0: ð1ÞThe integration of the equation over the interval [xj�1/2,xj+1/2] leads to

ð�ujÞt þf ðujþ1=2Þ � f ðuj�1=2Þ

Dx¼ 0; ð2Þ

where

�ujðtÞ ¼1

Dx

Z xjþ1=2

xj�1=2

uðx; tÞdx ð3Þ

and f(uj±1/2) are the numerical fluxes at the interfaces xj±1/2.

The determination of f(uj±1/2) can be split into the two stages. The first is a

projection (reconstruction) stage, in which the left-hand-side and right-hand-

side values of the state variables at the interfaces, uLj�1=2 and uRj�1=2, are com-

puted through the interpolation of �ui (i = j � l, . . ., j + r). The second stage is

an evolution (upwind) stage, in which the interface values of the numerical flux,f(uj±1/2), are computed from the given uLj�1=2 and uRj�1=2 by locally solving a Rie-

mann problem at the interfaces xj±1/2.

The focus of this paper is on the first stage. We will discuss how to com-

pute the left-hand-side and right-hand-side values of the state variables at the

interfaces, uLj�1=2 and uRj�1=2, by the two types of WENO interpolation. The

methods can be straightforwardly extended to the hyperbolic systems of con-

servation laws by applying the WENO interpolation to the characteristic

variables.

3. Upwind WENO5 scheme

For the sake of comparison, we will first give a brief review of the well-

known upwind WENO5 scheme developed by Liu et al. [14] and Jiang and

Shu [15].


Given a uniform five-point stencil from xj�2 to xj+2, following the ENO ap-

proach, there are three candidate stencils which can be used to construct an

essentially non-oscillatory third-order polynomial interpolation of uLjþ1=2. They

are from xj�2 to xj, and xj�1 to xj+1, and xj to xj+2, and the corresponding inter-

polations are

ðuLjþ1=2Þ0 ¼1

3�uj�2 �

7

6�uj�1 þ

11

6�uj;

ðuLjþ1=2Þ1 ¼ � 1

6�uj�1 þ

5

6�uj þ

1

3�ujþ1;

ðuLjþ1=2Þ2 ¼1

3�uj þ

5

6�ujþ1 �

1

6�ujþ2;

ð4Þ

respectively. A linear combination of these three lower-order polynomial inter-

polations gives a WENO interpolation of uLjþ1=2

uLjþ1=2 ¼xj

0

3�uj�2 �

7xj0 þ xj

1

6�uj�1 þ

11xj0 þ 5xj

1 þ 2xj2

6�uj

þ 2xj1 þ 5xj

2

6�ujþ1 �

xj2

6�ujþ2; ð5Þ

where xj0, x

j1, and xj

2 are the nonlinear weights defined as

xjk ¼ ajk

.X2

i¼0

aji and ajk ¼ Ck=ðeþ ISjkÞ

p: ð6Þ

Here e is a very small positive number, e.g., 10�6, to avoid the denominator

from zero, p is a free parameter, ISjk is the smoothness measurement of the solu-

tion over the kth candidate stencil, and Ck is the optimal linear weight for the

kth candidate stencil. According to [15], the optimal linear weights for the three

polynomial interpolations in (4) are C0 = 0.1, C1 = 0.6, and C2 = 0.3 respec-tively and the three smoothness indicators in (6) are

ISj0 ¼

1

4ð�uj�2 � 4�uj�1 þ 3�ujÞ2 þ

13

12ð�uj�2 � 2�uj�1 þ �ujÞ2;

ISj1 ¼

1

4ð�uj�1 � �ujþ1Þ2 þ

13

12ð�uj�1 � 2�uj þ �ujþ1Þ2;

ISj2 ¼

1

4ð3�uj � 4�ujþ1 þ �ujþ2Þ2 þ

13

12ð�uj � 2�ujþ1 þ �ujþ2Þ2:

ð7Þ

In the smooth regions, the nonlinear weights defined in (6) reduce to theoptimal linear weights and the WENO interpolation of uLjþ1=2 in (5) becomes

a fifth-order approximation to uLjþ1=2

uLjþ1=2 ¼2�uj�2 � 13�uj�1 þ 47�uj þ 27�ujþ1 � 3�ujþ2

60: ð8Þ


It is clear that the stencil used in (8) is upwind-biased with respect to the inter-

face xj+1/2. This is why we refer the resulting spatial discretization as the up-

wind WENO5 scheme.

Similarly, one can yield the corresponding WENO5 interpolation for uRjþ1=2

uRjþ1=2 ¼ �xjþ12

6�uj�1 þ

5xjþ12 þ 2xjþ1

1

6�uj þ

2xjþ12 þ 5xjþ1

1 þ 11xjþ10

6�ujþ1

� xjþ11 þ 7xjþ1

0

6�ujþ2 þ

xjþ10

3�ujþ3 ð9Þ

and its optimal linear version

uRjþ1=2 ¼�3�uj�1 þ 27�uj þ 47�ujþ1 � 13�ujþ2 þ 2�ujþ3

60: ð10Þ

It is found that both the stencil and coefficients for uRjþ1=2 in (9) and (10) are

different from those for uLjþ1=2 in (5) and (8). Therefore, even in the smooth re-

gions, a Riemann solver is still required at the interfaces. This is completely dif-

ferent from the central WENO scheme, which will be discussed in the next

section.

4. Central WENO4 scheme

Instead of directly approximating the interface values of the state variables

uLjþ1=2, using the same stencils and polynomial interpolations as those in (4), one

can also construct three second-order approximations to (ux)j. They are

½ðuxÞj�0 ¼uj�2 � 4uj�1 þ 3uj

2Dx;

½ðuxÞj�1 ¼�uj�1 þ ujþ1

2Dx;

½ðuxÞj�2 ¼�3uj þ 4ujþ1 � ujþ2

2Dx:

ð11Þ

The same linear combination of these three second-order approximations to(ux)j as the one for uLjþ1=2 in (5) yields a WENO approximation to (ux)j

ðuxÞj ¼xj

0uj�2þð�4xj0�xj

1Þuj�1þ3ðxj0�xj

2Þujþðxj1þ4xj

2Þujþ1�xj2ujþ2

2Dx:

ð12ÞHere xj

0, xj1, and xj

2 are the nonlinear weights same as those defined in (6) ex-cept that the optimal linear weights are different from those for uLjþ1=2. They are

C0 = 1/6, C1 = 2/3, and C2 = 1/6 here.

By contrast, the upwind WENO5 scheme presented in the last section results

in the following WENO approximation to (ux)j


ðuxÞj ¼ð2xj

0 þ xj2Þðuj�2 � ujþ2Þ � ð7xj

0 þ 3xj1 þ 5xj

2Þðuj�1 � ujþ1Þ6Dx

; ð13Þ

which can be considered as a linear combination of the following three second-

order approximations to (ux)j

½ðuxÞj�0 ¼2ðuj�2 � ujþ2Þ � 7ðuj�1 � ujþ1Þ

6Dx;

½ðuxÞj�1 ¼�uj�1 þ ujþ1

2Dx;

½ðuxÞj�2 ¼ðuj�2 � ujþ2Þ � 5ðuj�1 � ujþ1Þ

6Dx:

ð14Þ

However, when the nonlinear weights reduce to the optimal linear weights in

the smooth regions, both WENO approximations to (ux)j in (12) and (13) be-

come a fourth-order central approximation to (ux)j

ðuxÞj ¼ðuj�2 � ujþ2Þ � 8ðuj�1 � ujþ1Þ

12Dx: ð15Þ

It is clear that the WENO approximation to (ux)j in (13), resulting from the up-wind WENO5 scheme, looks much closer to the optimal linear version of (15)

in the form.

Following the work of Davis [20], we further apply (12) to approximate the

derivative of the primitive function U(x, t) (=R xx0uðn; tÞdn) at the interface xj+1/2.

According to the ‘‘Reconstruction via a Primitive Function’’ technique in [11],

the point values of the primitive function U(x, t) at the interfaces can be eval-

uated by

Ujþ1=2 ¼ DxXj

i¼i0

�ui: ð16Þ

Therefore, the application of (12) to (Ux)j+1/2 leads to a WENO interpolation

of uLjþ1=2

uLjþ1=2 ¼ ðUxÞjþ1=2

¼ �xj0�uj�1 þ ð3xj

0 þ xj1Þ�uj þ ðxj

1 þ 3xj2Þ�ujþ1 � xj

2�ujþ2

2: ð17Þ

Different from the upwind WENO5 interpolation of (5), the WENO interpola-

tion of (17) uses a symmetric stencil with respect to the interface xj+1/2. How-

ever, the interpolation is still upwind-biased because the nonlinear weights in

(17) are based on the same upwind-biased stencil as (5). On the other hand,

in the smooth regions, when the nonlinear weights in (17) reduce to the optimal

linear weights, the WENO interpolation of uLjþ1=2 in (17) becomes a fourth-

order central interpolation


uLjþ1=2 ¼��uj�1 þ 7�uj þ 7�ujþ1 � �ujþ2

12: ð18Þ

Here both the stencil and coefficients are symmetric with respect to the inter-

face xj+1/2. This is why we refer the interpolation of (17) to as a central WENO4

interpolation.The central WENO4 interpolation of uRjþ1=2 can be found in a similar way

uRjþ1=2 ¼�xjþ1

0 �uj�1 þ ð3xjþ10 þ xjþ1

1 Þ�uj þ ðxjþ11 þ 3xjþ1

2 Þ�ujþ1 � xjþ12 �ujþ2

2:

ð19ÞDifferent from their upwind WENO5 counterparts, the central WENO4 inter-

polation of uRjþ1=2 in (19) is almost the same as the central WENO4 interpola-

tion of uLjþ1=2 in (17) except that the nonlinear weights may change from node tonode. When the nonlinear weights further reduce to the optimal linear weights,

which are constants,

uRjþ1=2 ¼ uLjþ1=2: ð20Þ

Therefore, in the smooth regions, no Riemann problem exists at the interfaces.

The second evolution (upwind) stage in the Godunov-type schemes can be

saved. The resulting spatial discretization is a simple and efficient central

scheme.

The above central WENO4 scheme is conceptually very neat. It takes full

advantage of both a central scheme in the smooth regions and an upwind

scheme in the non-smooth regions. Based on the local smoothness measures

of the solution, the scheme achieves an automatic transition between centraland upwind schemes. The upwind bias contained in the above central WENO4

scheme is the minimum for a stable and numerically non-oscillatory scheme in

the sense that upwinding is only introduced through the nonlinear weights in

the non-smooth regions.

It is noteworthy that the central WENO4 interpolation of uLjþ1=2 in (17) is

derived from the central WENO4 differencing of (ux)j in (12). Therefore, the

smoothness measures used in (17) should be determined based on the three

third-order polynomials used for construction of those second-order approxi-mations to (ux)j in (11) rather than the following three polynomial interpola-

tions of uLjþ1=2

ðuLjþ1=2Þ0 ¼��uj�1 þ 3�uj

2;

ðuLjþ1=2Þ1 ¼�uj þ �ujþ1

2;

ðuLjþ1=2Þ2 ¼3�ujþ1 � �ujþ2

2:

ð21Þ


According to the definition of ISjk given in [15], the smoothness indicators for

those polynomials in (21) are

ISj0 ¼ ð�uj�1 � �ujÞ2;

ISj1 ¼ ð�uj � �ujþ1Þ2;

ISj2 ¼ ð�ujþ1 � �ujþ2Þ2:

ð22Þ

Unfortunately, the resulting hyperbolic solver is unstable because the nonlinear

weights are also based on a symmetric stencil. On the other hand, the smooth-

ness measures given in (7) lead to a stable central WENO4 scheme.

5. Numerical results and discussion

A central WENO scheme is conceptually very neat. It takes full advantage

of the two major categories of spatial discretization, central in the smooth

regions and upwind in the non-smooth regions. However, based on the

same multiple lower-order polynomials, the resulting central WENO scheme

is one order of accuracy lower than the resulting upwind WENO scheme.

In the following, we will investigate several typical test cases to com-

pare the accuracy between the upwind WENO5 scheme and the central

WENO4 scheme when they are combined with Roe�s approximate Riemannsolver.

The Trapezoidal scheme is used for time discretization because no numerical

dissipation exists in this scheme. A central WENO scheme has to solely rely on

the only upwinding flavor introduced through the nonlinear weights in the

non-smooth regions to achieve the stability and suppress the numerical

oscillations near a discontinuity. The implicit operator used is the LU-SGS

(lower–upper symmetric Gauss–Seidel) scheme with the spectral radius approx-

imation, which is used in a typical rotor Euler/Navier–Stokes code, TURNS

[21,22]. There is also an option of using Newton-type subiterations at each time

step for reduction of the linearization and factorization errors, and the

improvement of time accuracy. Four subiterations are used in the following

time-accurate computations.

5.1. Linear scalar convection

The first case considered is the numerical solution of the linear scalar con-vection equation

ut þ ux ¼ 0 ð06 x6 2Þ ð23Þwith the initial condition


uðx; 0Þ ¼

e� log 50 x�0:150:05ð Þ2 x0; 0:2;

1 0:3 < x < 0:4;

20x� 10 0:5 < x < 0:55;

12� 20x 0:556 x0; 0:6;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x�0:75

0:05

� �2q0:7 < x < 0:8;

0 otherwise;

8>>>>>>>>>><>>>>>>>>>>:

ð24Þ

which contains a smooth but narrow combination of a Gaussian function, asquare wave, a triangle wave, and a half ellipse, the same as in [23]. Each wave

spreads a distance of 10 cells on a coarser mesh of 200 cells and 20 cells on a

finer mesh of 400 cells. A CFL number of 0.4 is chosen for high time accuracy.

Fig. 1 presents the exact solution and the numerical results given by the

above two WENO schemes on the finer mesh after 500 time steps and on

the coarser mesh after 250 time steps. Here UWENO5 and CWENO4 stand

for Upwind WENO5 scheme presented in Section 3 and central WENO4

scheme presented in Section 4 respectively. It is found that although as ex-pected CWENO4 is slightly less accurate than UWENO5, the difference be-

tween the results given by these two methods is almost negligible, especially

for p = 1. The central WENO4 scheme seems more sensitive to the value of p

than the upwind WENO5 scheme. The difference between UWENO5 and

CWENO4 is slightly larger for p = 2 than for p = 1. Increasing p tends to damp

the peak values of the solution at those local extrema.

5.2. Sod’s shock-tube problem

The second selected case is the well-known Sod�s 1-D shock-tube problem

[24]. The governing equations are the 1-D Euler equations

o

ot

q

qu

e

0B@

1CAþ o

ox

qu

qu2 þ p

uðeþ pÞ

0B@

1CA ¼ 0; ð25Þ

and the initial data are

ðqL; uL; pLÞ ¼ ð1; 0; 1Þ; ðqR; uR; pRÞ ¼ ð0:125; 0; 0:1Þ: ð26ÞHere q, u, p and e are the density, velocity, pressure, and total energy per unit

volume respectively, and p = (c � 1)(e � qu2/2).Presented in Fig. 2 are the exact solution and the numerical results given by

the two WENO schemes on a uniform mesh of 100 cells over 0 6 x 6 1 after

125 time steps. For this case, it is found that the difference between the resultsgiven by UWENO5 and CWENO4 is negligible. The solutions are also not sen-

sitive to the choice of p.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.05 1.25 1.45 1.65 1.85

x

u

Exact

UWENO5

CWENO4

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.05 1.25 1.45 1.65 1.85

x

u

Exact

UWENO5

CWENO4

(a)

(b)

Fig. 1. Linear scalar convection (CFL = 0.4): (a) p = 1, N = 400, 500 time steps; (b) p = 2, N = 400,

500 time steps; (c) p = 1, N = 200, 250 time steps; (d) p = 2, N = 200, 250 time steps.


5.3. Woodward–Colella’s two interacting blast waves

The third selected case is Woodward–Colella�s problem in [25], which in-

volves the interaction of two blast waves. The initial data are the following:

(i) if 0 < x < 0.1, then

ðqL; uL; pLÞ ¼ ð1; 0; 1000Þ; ð27Þ(ii) if 0.1 < x < 0.9, then

ðqM; uM; pMÞ ¼ ð1; 0; 0:01Þ; ð28Þ

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.05 1.25 1.45 1.65 1.85

x

u

Exact

UWENO5

CWENO4

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.05 1.25 1.45 1.65 1.85

x

u

Exact

UWENO5

CWENO4

(c)

(d)

Fig. 1 (continued)


(iii) if 0.9 < x < 1, then

ðqR; uR; pRÞ ¼ ð1; 0; 100Þ; ð29Þwhich presents one shock at x = 0.1 and the other at x = 0.9. The boundaries

at x = 0 and x = 1 are solid walls with a reflective boundary condition. After

a certain time, these two shocks collide with each other. At the final time step

of t = 0.038, the flow field involves two shocks and three contact discon-

tinuities.

Presented in Fig. 3 are the numerical solutions produced by UWENO5 and

CWENO4 on a uniform mesh of 400 cells at t = 0.038. There is no exact solu-

tion for this case. It is found that except in the middle region after the left shock

and before the right contact discontinuity, both UWENO5 and CWENO4

schemes produce very similar results. In that particular region, however, the

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.4 0.7 1

x

ρ

0

0.2

0.4

0.6

0.8

1

1.2

ρ

Exact

UWENO5

CWENO4

0.1 0.4 0.7 1

x

Exact

UWENO5

CWENO4

(a)

(b)

Fig. 2. Density distribution of Sod�s shock-tube problem (N = 100, CFL = 0.4, 125 steps): (a) p = 1

and (b) p = 2.


results given by CWENO4 are not as smooth as those given by UWENO5,

especially in the distribution of the density and velocity for p = 2. While UWE-NO5 seems less sensitive to the choice of p, the results given by CWENO4 with

p = 1 look like smoother than those given by CWENO4 with p = 2 and are clo-

ser to the results given by UWENO5. Again, for the central WENO scheme,

increasing the value of p tends to reduce the peak values of the solutions at

the local extrama.

5.4. Shu–Osher’s problem

The fourth selected case is Shu–Osher�s problem in [13], which represents a

Mach 3 shock interacting with a sinusoidal density field. The initial data are

ðqL; uL; pLÞ ¼ ð3:86; 2:63; 10:33Þ; ðqR; uR; pRÞ ¼ ð1þ 0:2 sin 5x; 0; 1Þ:ð30Þ

0

1

2

3

4

5

6

0.4 0.5 0.6 0.7 0.8 0.9x

ρ

UWENO5CWENO4

0

1

2

3

4

5

6

0.4 0.5 0.6 0.7 0.8 0.9x

ρ

p=1 p=2

0.4 0.5 0.6 0.7 0.8 0.9x

0.4 0.5 0.6 0.7 0.8 0.9x

p=1 p=2

0

3

6

9

12

15

u

0

3

6

9

12

15

u

UWENO5CWENO4

(a)

(b)

(c)

0

50

100

150

200

250

300

350

400

450

0.4 0.5 0.6 0.7 0.8 0.9x

p

UWENO5CWENO4

0

50

100

150

200

250

300

350

400

450

0.4 0.5 0.6 0.7 0.8 0.9x

p

p=1 p=2

Fig. 3. Woodward–Colella�s blast waves problem (N = 400, Dt = 0.00005, t = 0.038): (a) density,

(b) velocity, and (c) pressure.


Similar to the first case, in this problem, there are several extrema in the

smooth regions which present a good case for examination of the accuracy

of a shock-capturing scheme.

Again, no exact solution exists for this problem. Fig. 4 presents the numer-

ical results given by the two WENO schemes on the uniform meshes of 200 and400 cells at t = 1.8. It is found that the difference between the results given by

UWENO5 and CWENO4 is almost negligible. Increasing the value of p tends

to smear out the peak values of the solutions at those local extrema.

0

1

2

3

4

5

-3 -1 1 3x

ρ

UWENO5

CWENO40

1

2

3

4

5

-3 -1 1 3x

ρ

p=1 p=2

0

1

2

3

4

5

-3 -1 1 3x

0

1

2

3

4

5

-3 -1 1 3x

ρ

UWENO5

CWENO4

p=1 p=2

UWENO5

CWENO4

UWENO5

CWENO4

ρ

(a)

(b)

Fig. 4. Density distribution of Shu–Osher�s problem (Dt = 0.002, t = 1.8): (a) N = 400 and (b)

N = 200.


6. Conclusions

There are two types of WENO schemes, upwind and central. A central

WENO scheme is conceptually very neat. It takes full advantage of the twomajor types of spatial discretization, central in the smooth regions and upwind

in the non-smooth regions. On the other hand, a central WENO scheme is one

order of accuracy lower than its upwind WENO counterpart. For most prob-

lems, the upwind and central WENO schemes produce very similar results.

However, for some tough cases, the solution given by the central WENO4

scheme is not as smooth as the one given by the upwind WENO5 scheme. A

central WENO scheme seems more sensitive to the choice of p. A better choice

for a central WENO scheme is p = 1.

Acknowledgements

This work is supported by NSF Grant DMI-0232255 and the technical mon-

itor is Dr. Juan E. Figueroa. The author would like to thank Prof. Chi-Wang

Shu for helpful discussions.


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Documents

Upwind and central WENO schemes