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Hermite WENO schemes and their application as limiters
for Runge–Kutta discontinuous Galerkin method:
one-dimensional case
Jianxian Qiu
a,1
, Chi-Wang Shu
b,*,2
a
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
b
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Received 16 May 2003; received in revised form 23 July 2003; accepted 23 July 2003
Abstract
In this paper, a class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite poly-
nomials, termed HWENO (Hermite WENO) schemes, for solving one-dimensional nonlinear hyperbolic conservation
law systems is presented. The construction of HWENO schemes is based on a finite volume formulation, Hermite
interpolation, and nonlinearly stable Runge–Kutta methods. The idea of the reconstruction in the HWENO schemes
comes from the original WENO schemes, however both the function and its first derivative values are evolved in time
and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes.
Comparing with the original WENO schemes of Liu et al. [J. Comput. Phys. 115 (1994) 200] and Jiang and Shu [J.
Comput. Phys. 126 (1996) 202], one major advantage of HWENO schemes is its compactness in the reconstruction. For
example, five points are needed in the stencil for a fifth-order WENO (WENO5) reconstruction, while only three points
are needed for a fifth-order HWENO (HWENO5) reconstruction. For this reason, the HWENO finite volume meth-
odology is more suitable to serve as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods, than the
original WENO finite volume methodology. Such applications in one space dimension is also developed in this paper.
Ó2003 Elsevier B.V. All rights reserved.
AMS: 65M06; 65M60; 65M99; 35L65
Keywords: WENO scheme; Hermite interpolation; High order accuracy; Runge–Kutta discontinuous Galerkin method; Limiters
www.elsevier.com/locate/jcp
Journal of Computational Physics 193 (2003) 115–135
*
Corresponding author. Tel.: +401-863-2549; fax: +401-863-1355.
E-mail addresses: jxqiu@ustc.edu.cn (J. Qiu), shu@dam.brown.edu (C.-W. Shu).
1
The research of this author is supported by NNSFC grant 10028103.
2
The research of this author is supported by NNSFC grant 10028103 while he is in residence at the Department of Mathematics,
University of Science and Technology of China, Hefei, Anhui 230026, PR China. Additional support is provided by ARO grant
DAAD19-00-1-0405 and NSF grant DMS-0207451.
0021-9991/$ - see front matter Ó2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.jcp.2003.07.026
1. Introduction
In this paper, we first construct a class of fifth-order weighted essentially non-oscillatory (WENO)
schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving
one-dimensional (1D) nonlinear hyperbolic conservation law systems
utþfðuÞx¼0;
uðx;0Þ¼u0ðxÞ:
ð1:1Þ
We then apply this HWENO finite volume methodology as limiters for the Runge–Kutta discontinuous
Galerkin (RKDG) methods. Only 1D case is considered in this paper. While the methodology can be
generalized in principle to multi dimensions, more work is needed to carry out the detailed design and this is
left for future research.
WENO schemes have been designed in recent years as a class of high order finite volume or finite dif-
ference schemes to solve hyperbolic conservation laws with the property of maintaining both uniform high
order accuracy and an essentially non-oscillatory shock transition. The first WENO scheme is constructed
in [19] for a third-order finite volume version in one space dimension. In [17], third and fifth-order finite
difference WENO schemes in multi space dimensions are constructed, with a general framework for the
design of the smoothness indicators and nonlinear weights. Finite difference WENO schemes of higher
orders (seventh to 11th order) are constructed in [1], and finite volume versions on unstructured and
structured meshes are designed in, e.g. [13,16,18,21,24]. WENO schemes are designed based on the suc-
cessful ENO schemes in [15,27,28]. Both ENO and WENO schemes use the idea of adaptive stencils in the
reconstruction procedure based on the local smoothness of the numerical solution to automatically achieve
high order accuracy and a non-oscillatory property near discontinuities. ENO uses just one (optimal in
some sense) out of many candidate stencils when doing the reconstruction; while WENO uses a convex
combination of all the candidate stencils, each being assigned a nonlinear weight which depends on the
local smoothness of the numerical solution based on that stencil. WENO improves upon ENO in ro-
bustness, better smoothness of fluxes, better steady state convergence, better provable convergence prop-
erties, and more efficiency. For a detailed review of ENO and WENO schemes, we refer to the lecture notes
[26].
The framework of the finite volume and finite difference WENO schemes is to evolve only one degree of
freedom per cell, namely the cell average for the finite volume version or the point value at the center of the
cell for the finite difference version. High order accuracy is achieved through a WENO reconstruction which
uses a stencil of kcells for kth order accuracy. Thus a fifth-order WENO scheme would need the infor-
mation from five neighboring cells in order to reconstruct the numerical flux. There are efforts in the lit-
erature to design schemes using a narrower stencil to achieve the same order of accuracy, through the
evolution of more than one degree of freedom per cell. For example, non-negativity, monotonicity or
convexity preserving cubic and quintic Hermite interpolation is discussed in [12]; various CIP type schemes
based on Hermite type interpolations are developed in, e.g. [20,29], and a second-order TVD scheme
satisfying all entropy conditions, based on evolving both the cell average and the slope per cell, is designed
in [3]. In the first part of this paper we follow this line of research and construct a class of fifth-order WENO
schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving the 1D
nonlinear hyperbolic conservation law systems (1.1). The construction of HWENO schemes is based on a
finite volume formulation, Hermite interpolation, and nonlinearly stable Runge–Kutta methods. The idea
of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the
function and its first derivative values are evolved in time and used in the reconstruction, while only the
function values are evolved and used in the original WENO schemes. Comparing with the original WENO
schemes of Liu et al. [19] and Jiang and Shu [17], one major advantage of HWENO schemes is its
116 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
compactness in the reconstruction. For example, five points are needed in the stencil for a fifth-order
WENO (WENO5) reconstruction, while only three points are needed for a fifth-order HWENO (HWE-
NO5) reconstruction.
The discontinuous Galerkin (DG) method can be considered as an extreme in the methodology de-
scribed above. It evolves kdegrees of freedom (in one dimension) per cell for a kth order accurate scheme,
thus no reconstruction is needed. The first DG method was introduced in 1973 by Reed and Hill [23], in the
framework of neutron transport (steady state linear hyperbolic equations). A major development of the DG
method was carried out by Cockburn et al. in a series of papers [5–9], in which they established a
framework to easily solve nonlinear time dependent hyperbolic conservation laws (1.1) using explicit,
nonlinearly stable high order Runge–Kutta time discretizations [27] and DG discretization in space with
exact or approximate Riemann solvers as interface fluxes and TVB (total variation bounded) limiter [25] to
achieve non-oscillatory properties for strong shocks. These schemes are termed Runge–Kutta discontin-
uous Galerkin (RKDG) methods. For a review of RKDG methods, see [10].
An important component of RKDG methods for solving conservation laws (1.1) with strong shocks in
the solutions is a nonlinear limiter, which is applied to control spurious oscillations. Although many
limiters exist in the literature, e.g. [2,4–9], they tend to degenerate accuracy when mistakenly used in smooth
regions of the solution. In [22], we initialized a study of using WENO methodology as limiters for RKDG
methods. The idea is to first identify ‘‘troubled cells’’, namely those cells where limiting might be needed,
then to abandon all moments in those cells except the cell averages and reconstruct those moments from the
information of neighboring cells using a WENO methodology. This technique works quite well in our one
and two-dimensional (2D) test problems [22]. However, one place in the approach of [22] which would
welcome improvements is that the reconstruction for the moments in troubled cells has to use the cell
average information from 2kþ1 neighboring cells, for (kþ1)th order RKDG methods of piecewise
polynomials of degree k. This stencil is significantly wider than the original RKDG methodology. For this
reason, the HWENO finite volume method developed in this paper is more suitable to serve as limiters for
the RKDG methods, since it uses much fewer neighboring cells to obtain a reconstruction of the same order
of accuracy. Such applications in one space dimension is also developed in this paper.
The organization of this paper is as follows. In Section 2, we describe in detail the construction and
implementation of HWENO schemes with Runge–Kutta time discretizations, for 1D scalar and system
equations (1.1). In Section 3, we investigate the usage of the HWENO finite volume methodology as
limiters for RKDG methods, following the idea in [22], with the goal of obtaining a robust and high order
limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock
transition for RKDG methods. In Section 4 we provide extensive numerical examples to demonstrate the
behavior of the HWENO schemes and DG methods with HWENO limiters with Runge–Kutta time dis-
cretizations. Concluding remarks are given in Section 5.
2. The construction of Hermite WENO schemes
In this section we first consider 1D scalar conservation laws (1.1). For simplicity, we assume that the grid
points fxigare uniformly distributed with the cell size xiþ1xi¼Dxand cell centers xiþ1=2¼1
2ðxiþxiþ1Þ.
We also denote the cells by Ii¼½xi1=2;xiþ1=2.
Let v¼uxand gðu;vÞ¼f0ðuÞux¼f0ðuÞv. From (1.1) and its spatial derivative we obtain
utþfðuÞx¼0;uðx;0Þ¼u0ðxÞ;
vtþgðu;vÞx¼0;vðx;0Þ¼v0ðxÞ:
ð2:1Þ
We denote the cell averages of uand vas
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 117
uiðtÞ¼ 1
DxZIi
uðx;tÞdx;viðtÞ¼ 1
DxZIi
vðx;tÞdx:
Integrating (2.1) over the cell Iiwe obtain an equivalent formulation of the conservation laws
duiðtÞ
dt¼1
Dxðfðuðxiþ1=2;tÞÞ fðuðxi1=2;tÞÞÞ;
dviðtÞ
dt¼1
Dxðgðuðxiþ1=2;tÞ;vðxiþ1=2;tÞÞ gðuðxi1=2;tÞ;vðxi1=2;tÞÞÞ:
(ð2:2Þ
We approximate (2.2) by the following conservative scheme
duiðtÞ
dt¼1
Dxð^
ffiþ1=2^
ffi1=2Þ;
dviðtÞ
dt¼1
Dxð^
ggiþ1=2^
ggi1=2Þ;
(ð2:3Þ
where the numerical fluxes ^
ffiþ1=2and ^
ggiþ1=2are defined by:
^
ffiþ1=2¼hu
iþ1=2;uþ
iþ1=2
;
^
ggiþ1=2¼Hu
iþ1=2;uþ
iþ1=2;v
iþ1=2;vþ
iþ1=2
;ð2:4Þ
where u
iþ1=2and v
iþ1=2are numerical approximations to the point values of uðxiþ1=2;tÞand vðxiþ1=2;tÞre-
spectively from left and right. The fluxes in (2.4) are subject to the usual conditions for numerical fluxes,
such as Lipschitz continuity and consistency with the physical fluxes fðuÞand gðu;vÞ.
In this paper we use the following local Lax–Friedrichs fluxes:
hða;bÞ¼1
2½fðaÞþfðbÞaðbaÞ;
Hða;b;c;dÞ¼1
2½gða;cÞþgðb;dÞaðdcÞ;
ð2:5Þ
where a¼maxu2Djf0ðuÞj, with D¼½minða;bÞ;maxða;bÞ.
The method of lines ODE (2.3) is then discretized in time by a TVD Runge–Kutta method in [27]. The
third-order version in [27] is used in this paper.
The first-order ‘‘building block’’ of this scheme can be obtained by using the cell averages uiand vito
replace the point values u
iþ1=2,uþ
i1=2and v
iþ1=2,vþ
i1=2respectively, and using Euler forward for the time
discretization. The result is the following scheme
unþ1
i¼un
ikhðun
i;un
iþ1Þhðun
i1;un
iÞ
;
vnþ1
i¼vn
ikH un
i;un
iþ1;vn
i;vn
iþ1
Hun
i1;un
i;vn
i1;vn
i
ð2:6Þ
with the numerical fluxes hand Hdefined by (2.5). Here k¼Dt=Dx. For this building block we have the
following total variation stability result, where for simplicity we assume ais a constant:
Proposition 2.1. The scheme (2.6), with the numerical fluxes hand Hdefined by (2.5) and under the CFL
condition ka 61;satisfies
TV ðunþ1Þ6TV ðunÞ;jjvnþ1jjL16jjvnjjL1;
where the norms are defined by
TV ðuÞX
i
juiþ1uij;jjvjjL1X
i
jvijDx:
118 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
Proof. The total variation diminishing property TV ðunþ1Þ6TV ðunÞis a consequence of the monotone
scheme satisfied by u, see for example [11]. We thus prove only the L1stability result for v. By using def-
initions of the scheme (2.6) and the numerical fluxes (2.5) we obtain
1
Dxjjvnþ1jjL1¼X
i
un
i
k
2f0un
iþ1
vn
iþ1
f0un
i1
vn
i1avn
iþ1
2vn
iþvn
i1
¼X
i
ð1
kaÞvn
iþk
2a
f0un
iþ1
vn
iþ1þk
2a
þf0un
i1
vn
i1
6X
i
ð1kaÞvn
i
þX
i
k
2a
f0un
i
vn
i
þX
i
k
2a
þf0un
i
vn
i
¼X
i
vn
i
¼1
DxjjvnjjL1:
Here, we have used the definition of aand the CFL condition ka 61 in the inequality above and have
ignored boundary terms by assuming periodic or compact boundary conditions.
Since vapproximates the derivative of u, the L1norm of vis equivalent to the total variation norm of u.
Thus the proposition indicates that the base first-order scheme is TVD, both in a direct measurement of the
total variation norm of uand in an indirect measurement of the total variation norm of uthrough the L1
norm of v. This gives us a solid foundation to build higher order schemes using this building block.
The key component of the HWENO schemes is the reconstruction, from the cell averages fui;vigto
the points values fu
iþ1=2;v
iþ1=2g. This reconstruction should be both high order accurate and essentially
non-oscillatory. We outline the procedure of this reconstruction for the fifth-order accuracy case in the
following.
Step 1. Reconstruction of fu
iþ1=2gby HWENO from the cell averages fui;vig.
1. Given the small stencils S0¼fIi1;Iig,S1¼fIi;Iiþ1gand the bigger stencil T¼fS0;S1g, we construct
Hermite quadratic reconstruction polynomials p0ðxÞ;p1ðxÞ;p2ðxÞand a fourth-degree reconstruction poly-
nomial qðxÞsuch that:
1
DxZIiþj
p0ðxÞdx¼uiþj;j¼1;0;1
DxZIi1
p0
0ðxÞdx¼vi1;
1
DxZIiþj
p1ðxÞdx¼uiþj;j¼0;1;1
DxZIiþ1
p0
1ðxÞdx¼viþ1;
1
DxZIiþj
p2ðxÞdx¼uiþj;j¼1;0;1:
1
DxZIiþj
qðxÞdx¼uiþj;j¼1;0;1;1
DxZIiþj
q0ðxÞdx¼viþj;j¼1;1:
In fact, we only need the values of these polynomials at the cell boundary xiþ1=2given in terms of the cell
averages, which have the following expressions:
p0ðxiþ1=2Þ¼7
6ui1þ13
6ui2Dx
3vi1;
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 119
p1ðxiþ1=2Þ¼1
6uiþ5
6uiþ1Dx
3viþ1;
p2ðxiþ1=2Þ¼1
6ui1þ5
6uiþ1
3uiþ1;
qðxiþ1=2Þ¼ 23
120 ui1þ19
30 uiþ67
120 uiþ1Dx3
40 vi1
þ7
40 viþ1:
2. We find the combination coefficients, also called linear weights, denoted by c0,c1and c2, satisfying:
qðxiþ1=2Þ¼X
2
j¼0
cjpjðxiþ1=2Þ
for all the cell averages of uand vin the bigger stencil T. This leads to
c0¼9
80 ;c1¼21
40 ;c2¼29
80 :
3. We compute the smoothness indicator, denoted by bj, for each stencil Sj, which measures how smooth
the function pjðxÞis in the target cell Ii. The smaller this smoothness indicator bj, the smoother the function
pjðxÞis in the target cell. We use the same recipe for the smoothness indicator as in [17]
bj¼X
2
l¼1ZIi
Dx2l1ol
oxlpjðxÞ
2
dx:ð2:7Þ
In the actual numerical implementation the smoothness indicators bjare written out explicitly as quadratic
forms of the cell averages of uand vin the stencil:
b0¼ð2ui1þ2uiDxvi1Þ2þ13
3ðui1þuiDxvi1Þ2;
b1¼ð2uiþ2uiþ1Dxviþ1Þ2þ13
3ðuiþuiþ1þDxviþ1Þ2;
b2¼1
4ðui1þuiþ1Þ2þ13
12 ðui1þ2uiuiþ1Þ2:
4. We compute the nonlinear weights based on the smoothness indicators
xj¼xj
Pkxk
;xk¼ck
ðeþbkÞ2;ð2:8Þ
where ckare the linear weights determined in Step 1.2 above, and eis a small number to avoid the de-
nominator to become 0. We are using e¼106in all the computation in this paper. The final HWENO
reconstruction is then given by
u
iþ1=2X
2
j¼0
xjpjðxiþ1=2Þ:ð2:9Þ
The reconstruction to uþ
i1=2is mirror symmetric with respect to xiof the above procedure.
120 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
Step 2. Reconstruction of the derivative values fv
iþ1=2gby HWENO from the cell averages fui;vig.
5. Given the small stencils S0¼fIi1;Iig,S1¼fIi;Iiþ1gand the bigger stencil T¼fS0;S1g, we construct
Hermite cubic reconstruction polynomials p0ðxÞ;p1ðxÞ;p2ðxÞand a fifth-degree reconstruction polynomial
qðxÞsuch that:
1
DxZIiþj
p0ðxÞdx¼uiþj;1
DxZIiþj
p0
0ðxÞdx¼viþj;j¼1;0;
1
DxZIiþj
p1ðxÞdx¼uiþj;1
DxZIiþj
p0
1ðxÞdx¼viþj;j¼0;1;
1
DxZIiþj
p2ðxÞdx¼uiþj;j¼1;0;1;1
DxZIi
p0
2ðxÞdx¼vi;
1
DxZIiþj
qðxÞdx¼uiþj;1
DxZIiþj
q0ðxÞdx¼viþj;j¼1;0;1:
In fact, we only need the values of the derivative of these polynomials at the cell boundary xiþ1=2given in
terms of the cell averages, which have the following expressions:
p0
0ðxiþ1=2Þ¼ 4
Dxðui1uiÞþ3
2vi1þ7
2vi;
p0
1ðxiþ1=2Þ¼ 2
Dxðui1þuiÞ1
2vi1
2viþ1;
p0
2ðxiþ1=2Þ¼ 1
4Dxui1
ð4uiþ3uiþ1Þþ1
2vi;
q0ðxiþ1=2Þ¼ 1
Dx
1
4ui1
2uiþ7
4uiþ1þ1
12 vi11
6vi5
12 viþ1:
6. Compute linear weights c0
0,c0
1and c0
2, satisfying
q0ðxiþ1=2Þ¼X
2
j¼0
c0
jp0
jðxiþ1=2Þ
for all the cell averages of uand vin the bigger stencil T. This leads to
c0
0¼1
18 ;c0
1¼5
6;c0
2¼1
9:
7. We define the smoothness indicators for the reconstruction of derivatives as
bj¼X
3
l¼2ZIi
ðDxÞð2l1Þol
oxlpjðxÞ
2
dx:ð2:10Þ
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 121
Notice that the summation starts from the second derivative rather than from the first, as we are now
reconstructing the first derivative rather than the function values. We can again write these smoothness
indicators out explicitly as quadratic forms of the cell averages of uand vin the stencil:
b0¼4ð3ðui1uiÞþDxðvi1þ2viÞÞ2þ39
4ð2ðui1uiÞþDxðvi1þviÞÞ2;
b1¼4ð3ðuiuiþ1ÞDxð2viþviþ1ÞÞ2þ39
4ð2ðuiuiþ1ÞþDxðviþviþ1ÞÞ2;
b2¼ðui12uiþuiþ1Þ2þ39
16 ðuiþ1ui12DxviÞ2:
8. We compute the nonlinear weights by
xj¼xj
Pkxk
;xk¼c0
k
ðeþbkÞ2:ð2:11Þ
The final HWENO reconstruction to v
iþ1=2is then given by
v
iþ1=2X
2
j¼0
xjp0
jðxiþ1=2Þ:ð2:12Þ
The reconstruction to vþ
i1=2is mirror symmetric with respect to xiof the above procedure.
We remark that a more natural procedure would have been using the same small stencils and lower order
polynomials in both Step 1 and Step 2 above, which would have saved computational time as the costly
smoothness indicators would have to be computed only once. Unfortunately this does not work as suitable
linear weights do not exist in Step 2.2 above for such choices.
For systems of conservation laws, such as the Euler equations of gas dynamics, both of the recon-
structions from fui;vigto fu
iþ1=2gand fv
iþ1=2gare performed in the local characteristic directions to avoid
oscillation. For details of such local characteristic decompositions, see, e.g. [26].
3. HWENO reconstruction as limiters for the discontinuous Galerkin method
In [22], we have started the study of using WENO reconstruction methodology as limiters for the RKDG
methods. The first step in the procedure is to identify the ‘‘troubled cells’’, namely those cells which might
need the limiting procedure. In [22] as well as in this paper, we use the usual minmod type TVB limiters as in
[5,7,9]. That is, whenever the minmod limiter changes the slope, the cell is declared to be a troubled cell.
This identification of troubled cells is not optimal. Often smooth cells, especially those near smooth ex-
trema, are mistakenly identified as troubled cells. However, the idea of using WENO reconstructions in
those cells is to maintain high order accuracy even if smooth cells are mistaken as troubled cells. The second
step is to replace the solution polynomials in the troubled cells by reconstructed polynomials which
maintain the original cell averages (for conservation), have the same order of accuracy as before, but are
less oscillatory. In [22], regular finite volume type WENO reconstruction based on cell averages of
neighbors is used for the second step. In this section, we apply the HWENO reconstruction procedure
developed in the previous section for the second step, which reduces the stencil of the reconstruction while
maintaining the same high order accuracy.
122 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
The DG solution as well as the test function space is given by Vk
h¼fp:pjIi2PkðIiÞg, where PkðIiÞis the
space of polynomials of degree 6kon the cell Ii. We adopt a local orthogonal basis over Ii,
fvðiÞ
lðxÞ;l¼0;1;...;kg, namely the scaled Legendre polynomials
vðiÞ
0ðxÞ¼1;vðiÞ
1ðxÞ¼xxi
Dxi
;vðiÞ
2ðxÞ¼ xxi
Dxi
2
1
12 ;...
Then the numerical solution uhðx;tÞin the space Vk
hcan be written as
uhðx;tÞ¼X
k
l¼0
uðlÞ
iðtÞvðiÞ
lðxÞfor x2Iið3:1Þ
and the degrees of freedom uðlÞ
iðtÞare the moments defined by
uðlÞ
iðtÞ¼ 1
alZIi
uhðx;tÞvðiÞ
lðxÞdx;l¼0;1;...;k;
where al¼RIiðvðiÞ
lðxÞÞ2dxare the normalization constants since the basis is not orthonormal. In order to
determine the approximate solution, we evolve the degrees of freedom uðlÞ
i
d
dtuðlÞ
iþ1
alZIi
fðuhðx;tÞÞ d
dxvðiÞ
lðxÞdxþ^
ffu
iþ1=2;uþ
iþ1=2
vðiÞ
lðxiþ1=2Þ
^
ffu
i1=2;u
iþ1=2
vðiÞ
lðxi1=2Þ¼0;l¼0;1;...;k;ð3:2Þ
where u
iþ1=2¼uhðx
iþ1=2;tÞare the left and right limits of the discontinuous solution uhat the cell interface
xiþ1=2,^
ff ðu;uþÞis a monotone flux (non-decreasing in the first argument and non-increasing in the second
argument) for the scalar case and an exact or approximate Riemann solver for the system case. The
semidiscrete scheme (3.2) is discretized in time by a nonlinearly stable Runge–Kutta time discretization, e.g.
the third-order version in [27]. The integral term in (3.2) can be computed either exactly or by a suitable
numerical quadrature accurate to at least OðDxkþlþ2Þ.
The limiter adopted in [7] is described below in some detail, as it is the one used in [22] and in this paper
to detect ‘‘troubled cells’’. Denote
u
iþ1=2¼uð0Þ
iþ~
uui;uþ
i1=2¼uð0Þ
i~
~
uu
~
uui
From (3.1) we can see that
~
uui¼X
k
l¼1
uðlÞ
ivðiÞ
lðxiþ1=2Þ;~
~
uu
~
uui¼X
k
l¼1
uðlÞ
ivðiÞ
lðxi1=2Þ:
These are modified by either the standard minmod limiter [14]
~
uuðmodÞ
i¼m~
uui;Dþuð0Þ
i;Duð0Þ
i
;~
~
uu
~
uuðmodÞ
i¼mð~
~
uu
~
uui;Dþuð0Þ
i;Duð0Þ
iÞ;
where mis given by
mða1;a2;...;anÞ¼ smin16j6njajjif signða1Þ¼signða2Þ¼ ¼signðanÞ¼s;
0 otherwise
ð3:3Þ
or by the TVB modified minmod function [25]
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 123
~
mmða1;a2;...;anÞ¼ a1if ja1j6MDx2;
mða1;a2;...;anÞotherwise;
ð3:4Þ
where M>0 is a constant. The choice of Mdepends on the solution of the problem. For scalar problems it
is possible to estimate Mby the initial condition as in [7] (Mis proportional to the second derivative of the
initial condition at smooth extrema), however it is more difficult to estimate Mfor the system case. If Mis
chosen too small, accuracy may degenerate at smooth extrema of the solution; however if Mis chosen too
large, oscillations will appear.
In [22] and in this paper we use the limiter described above to identify ‘‘troubled cells’’, namely, if one of
the minmod functions gets enacted (returns other than the first argument), this cell is declared ‘‘troubled’’
and marked for further reconstructions. Since the HWENO reconstruction maintains the high order ac-
curacy in the troubled cells, it is less crucial to choose an accurate M. We present in Section 3 numerical
results obtained with different MÕs. Basically, if Mis chosen too small, more good cells will be declared as
troubled cells and will be subject to unnecessary HWENO reconstructions. This does increase the com-
putational cost but does not degrade the order of accuracy in these cells.
For the troubled cells, we would like to reconstruct the polynomial solution while retaining its cell
average. In other words, we will reconstruct the degrees of freedom, or the moments, uðlÞ
ifor the troubled
cell Iifor l¼1;...;kand retain only the cell average uð0Þ
i.
For the third-order k¼2 case, we summarize the procedure to reconstruct the first and second moments
uð1Þ
iand uð2Þ
ifor a troubled cell Iiusing HWENO:
Step 1. Reconstruction of the first moment uð1Þ
iby HWENO.
1. Given the small stencils S0¼fIi1;Iig,S1¼fIi;Iiþ1gand the bigger stencil T¼fS0;S1g, we construct
Hermite quadratic reconstruction polynomials p0ðxÞ;p1ðxÞ;p2ðxÞand a fourth-degree reconstruction poly-
nomial qðxÞsuch that:
ZIiþj
p0ðxÞdx¼uð0Þ
iþja0;j¼1;0;ZIi1
p0ðxÞvði1Þ
1ðxÞdx¼uð1Þ
i1a1;
ZIiþj
p1ðxÞdx¼uð0Þ
iþja0;j¼0;1;ZIiþ1
p1ðxÞvðiþ1Þ
1ðxÞdx¼uð1Þ
iþ1a1;
ZIiþj
p2ðxÞdx¼uð0Þ
iþja0;j¼1;0;1;
ZIiþj
qðxÞdx¼uð0Þ
iþja0;j¼1;0;1;ZIiþj
qðxÞvðiþjÞ
1ðxÞdx¼uð1Þ
iþja1;j¼1;1:
We now obtain:
ZIi
p0ðxÞvðiÞ
1ðxÞdx¼a12uð0Þ
i1þ2uð0Þ
iuð1Þ
i1;
ZIi
p1ðxÞvðiÞ
1ðxÞdx¼a12uð0Þ
iþ2uð0Þ
iþ1uð1Þ
iþ1;
ZIi
p2ðxÞvðiÞ
1ðxÞdx¼a1uð0Þ
i1þuð0Þ
iþ1.2;
124 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
ZIi
qðxÞvðiÞ
1ðxÞdx¼al
15
19 uð0Þ
i1
uð0Þ
iþ111
38 uð1Þ
i1
þuð1Þ
iþ1:
2. We find the combination coefficients, also called linear weights, denoted by c0,c1and c2, satisfying:
ZIi
qðxÞvðiÞ
1ðxÞdx¼X
2
j¼0
cjZIi
pjðxÞvðiÞ
1ðxÞdx;
which leads to
c0¼11
38 ;c1¼11
38 ;c2¼8
19 :
3. We compute the smoothness indicator bjby (2.7), and the nonlinear weights based on the smoothness
indicators by (2.8). The first moment of the reconstructed polynomial is then given by
uð1Þ
i¼1
a1X
2
j¼0
xjZIi
pjðxÞvðiÞ
1ðxÞdx:ð3:5Þ
Step 2. Reconstruction of the second moment uð2Þ
iby HWENO. When the first moment uð1Þ
iis needed we
use the reconstructed one from Step 1.
4. Given the small stencils S0¼fIi1;Iig,S1¼fIi;Iiþ1gand the bigger stencil T¼fS0;S1g, we construct
Hermite cubic reconstruction polynomials p0ðxÞ;p1ðxÞ;p2ðxÞand a fifth-degree reconstruction polynomial
qðxÞsuch that:
ZIiþj
p0ðxÞdx¼uð0Þ
iþja0;ZIiþj
p0ðxÞvðiþjÞ
1ðxÞdx¼uð1Þ
iþja1;j¼1;0;
ZIiþj
p1ðxÞdx¼uð0Þ
iþja0;ZIiþj
p1ðxÞvðiþjÞ
1ðxÞdx¼uð1Þ
iþja1;j¼0;1;
ZIiþj
p2ðxÞdx¼uð0Þ
iþja0;j¼1;0;1;ZIi
p2ðxÞvðiÞ
1dx¼uð1Þ
ia1;
ZIiþj
qðxÞdx¼uð0Þ
iþja0;ZIiþj
qðxÞvðiþjÞ
1ðxÞdx¼uð1Þ
iþja1;j¼1;0;1;
which lead to
ZIi
p0ðxÞvðiÞ
2ðxÞdx¼a2
15
4uð0Þ
i1
15
4uð0Þ
iþ11
8uð1Þ
i1þ19
8uð1Þ
i;
ZIi
p1ðxÞvðiÞ
2ðxÞdx¼a215
4uð0Þ
iþ15
4uð0Þ
iþ119
8uð1Þ
i11
8uð1Þ
iþ1;
ZIi
p2ðxÞvðiÞ
2ðxÞdx¼a2
1
2uð0Þ
i1
uð0Þ
iþ1
2uð0Þ
iþ1;
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 125
ZIi
qðxÞvðiÞ
2dx¼a2
73
56 uð0Þ
i1
73
28 uð0Þ
iþ73
56 uð0Þ
iþ1þ45
112 uð1Þ
i145
112 uð1Þ
iþ1:
5. We find the linear weights denoted by c0,c1and c2satisfying
ZIi
qðxÞvðiÞ
2dxðxÞ¼X
2
j¼0
cjZIi
pjðxÞvðiÞ
2ðxÞdx;
which leads to
c0¼45
154 ;c1¼45
154 ;c2¼32
77 :
6. We compute the smoothness indicator bjby (2.10). The nonlinear weights are then computed based on
the smoothness indicators using (2.8). The second moment of the reconstructed polynomial is then given by
uð2Þ
i¼1
a2X
2
j¼0
xjZIi
pjðxÞvðiÞ
2ðxÞdx:ð3:6Þ
4. Numerical results
In this section we present the results of our numerical experiments for the fifth-order HWENO schemes
with the third-order TVD Runge–Kutta method (HWENO5-RK3) and the third-order DG method with
HWENO limiter (DG3-HWENO5-RK3) developed in the previous sections, and compare them with the
fifth-order finite volume WENO schemes in [26] and DG3 with TVB limiter [7]. A uniform mesh with N
cells is used for all the test cases, the CFL number is taken as 0.8 for both HWENO5 and WENO5, and
0.18 for DG3-HWENO5-RK3 except for some accuracy tests where a suitably reduced time step is used to
guarantee that spatial error dominates.
4.1. Accuracy tests
We first test the accuracy of the schemes on nonlinear scalar problems and nonlinear systems. In the
accuracy tests the TVB constant Mis taken as 0.01 (very close to a TVD limiter) for identifying the troubled
cells in order to test the effect of the HWENO reconstruction for wrongly identified troubled cells in smooth
regions.
Example 4.1. We solve the following nonlinear scalar Burgers equation
utþu2
2
x
¼0ð4:1Þ
with the initial condition uðx;0Þ¼0:5þsinðpxÞ, and a 2-periodic boundary condition. When t¼0:5=pthe
solution is still smooth, and the errors and numerical orders of accuracy by the HWENO5-RK3 scheme
and by the WENO5-RK3 scheme [26] are shown in Table 1. We can see that both HWENO5-RK3 and
WENO5-RK3 schemes achieve their designed order of accuracy, and HWENO5-RK3 produces smaller
126 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
errors than WENO5-RK3 for the same mesh. Notice, however, that HWENO5-RK3 is almost twice as
costly as WENO5-RK3 for the same mesh, as two reconstructions are involved for HWENO instead of just
one for WENO. The results for DG3-HWENO5-RK3 and DG3-RK3 with no limiter are shown in Table 2.
We can see that both schemes achieve their designed order of accuracy with comparable errors for the same
mesh.
Example 4.2. We solve the following nonlinear system of Euler equations
utþfðuÞx¼0ð4:2Þ
with
u¼ðq;qv;EÞT;fðuÞ¼ðqv;qv2þp;vðEþpÞÞT:
Here qis the density, vis the velocity, Eis the total energy, pis the pressure, which is related to the total
energy by E¼p=c1þ1=2qv2with c¼1:4. The initial condition is set to be qðx;0Þ¼1þ0:2 sinðpxÞ,
vðx;0Þ¼1, pðx;0Þ¼1, with a 2-periodic boundary condition. The exact solution is qðx;tÞ¼1þ
0:2 sinðpðxtÞÞ,vðx;tÞ¼1, pðx;tÞ¼1. We compute the solution up to t¼2. The errors and numerical
orders of accuracy of the density qfor the HWENO5-RK3 scheme are shown in Table 3, in comparison
with the results of WENO5-RK3 in [26]. We can see that both schemes achieve their designed order of
accuracy, and HWENO5-RK3 is more accurate than WENO5-RK3 on the same mesh. The results for
DG3-HWENO5-RK3 and DG3-RK3 with no limiter are shown in Table 4. We can see that both schemes
achieve their designed order of accuracy, however DG3-HWENO5-RK3 has larger errors for the same
mesh.
Table 2
Burgers equation utþðu2=2Þx¼0 with initial condition uðx;0Þ¼0:5þsinðpxÞ
NDG with HWENO limiter DG with no limiter
L1error Order L1error Order L1error Order L1error Order
10 1.41E )02 8.09E )02 3.35E )03 2.21E )02
20 1.12E )03 3.66 7.09E )03 3.51 4.00E )04 3.07 3.59E )03 2.62
40 7.99E )05 3.81 5.78E )04 3.62 5.11E )05 2.97 5.78E )04 2.64
80 8.34E )06 3.26 8.26E )05 2.81 6.46E )06 2.98 8.26E )05 2.81
160 9.97E )07 3.06 1.14E )05 2.86 8.14E )07 2.99 1.14E )05 2.86
320 1.22E )07 3.03 1.50E )06 2.92 1.02E )07 2.99 1.50E )06 2.92
DG3-HWENO5-RK3 and DG3-RK3 with no limiters. t¼0:5=p.L1and L1errors. Uniform meshes with Ncells.
Table 1
Burgers equation utþðu2=2Þx¼0 with initial condition uðx;0Þ¼0:5þsinðpxÞ
NHWENO5-RK3 WENO5-RK3
L1error Order L1error Order L1error Order L1error Order
10 5.06E )03 1.79E)02 8.42E )03 2.67E )02
20 4.93E )04 3.36 3.33E)03 2.43 1.04E )03 3.02 7.09E )03 1.91
40 3.65E )05 3.76 3.24E)04 3.36 8.86E )05 3.55 7.47E )04 3.25
80 1.61E )06 4.51 1.51E)05 4.43 4.17E )06 4.41 4.09E )05 4.19
160 6.25E )08 4.68 5.49E)07 4.78 1.67E )07 4.64 1.44E )06 4.82
320 1.86E )09 5.07 2.06E)08 4.74 5.14E )09 5.02 4.66E )08 4.95
HWENO5-RK3 and WENO5-RK3. t¼0:5=p.L1and L1errors. Uniform meshes with Ncells.
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 127
4.2. Test cases with shocks
Example 4.3. We solve the same nonlinear Burgers equation (4.1) as in Example 4.1 with the same initial
condition uðx;0Þ¼0:5þsinðpxÞ, except that we now plot the results at t¼1:5=pwhen a shock has already
appeared in the solution. In Fig. 1, the solutions of HWENO5-RK3 (left) and DG3-HWENO5-RK3 (right)
with N¼80 cells are shown. The solid line is the exact solution. We can see that both schemes give non-
oscillatory shock transitions for this problem.
Example 4.4. We solve the nonlinear non-convex scalar Buckley–Leverett problem
utþ4u2
4u2þð1uÞ2
!
x
¼0ð4:3Þ
with the initial data u¼1 when 1
26x60 and u¼0 elsewhere. The solution is computed up to t¼0:4.
The exact solution is a shock-rarefaction-contact discontinuity mixture. We remark that some high order
schemes may fail to converge to the correct entropy solution for this problem. In Fig. 2, the solutions of
HWENO5-RK3 (left) and DG3-HWENO5-RK3 (right) with N¼80 cells are shown. The solid line is the
exact solution. We can see that both schemes give good resolutions to the correct entropy solution for this
problem.
Example 4.5. We solve the Euler equations (4.2) with a Riemann initial condition for the Lax Problem
ðq;v;pÞ¼ð0:445;0:698;3:528Þfor x60;ðq;v;pÞ¼ð0:5;0;0:571Þfor x>0:
Table 3
Euler equations. qðx;0Þ¼1þ0:2 sinðpxÞ,vðx;0Þ¼1, pðx;0Þ¼1
NHWENO5-RK3 WENO5-RK3
L1error Order L1error Order L1error Order L1error Order
10 3.33E )03 5.10E )03 6.63E )03 5.92E )03
20 1.28E )04 4.70 2.43E )04 4.39 3.04E )04 4.45 2.91E )04 4.35
40 3.82E )06 5.07 7.34E )06 5.05 9.08E )06 5.06 9.19E )06 4.99
80 1.17E )07 5.02 2.30E )07 5.00 2.80E )07 5.02 2.95E )07 4.96
160 3.62E )09 5.02 6.54E )09 5.14 8.72E )09 5.00 8.67E )09 5.09
320 1.08E )10 5.06 1.85E )10 5.14 2.71E )10 5.01 2.45E )10 5.15
HWENO5-RK3 and WENO5-RK3 using Nequally spaced cells. t¼2. L1and L1errors of density q.
Table 4
Euler equations. qðx;0Þ¼1þ0:2 sinðpxÞ,vðx;0Þ¼1, pðx;0Þ¼1
NDG with HWENO limiter DG with no limiter
L1error Order L1error Order L1error Order L1error Order
10 2.79E)03 4.51E )03 1.41E )05 2.17E )05
20 1.05E)04 4.73 4.46E )04 3.34 8.14E)07 4.11 1.29E )06 4.08
40 2.31E)05 2.18 4.51E )05 3.31 7.06E)08 3.53 1.11E )07 3.54
80 3.27E)06 2.82 5.36E )06 3.07 7.84E)09 3.17 1.23E )08 3.17
160 4.21E)07 2.96 6.98E )07 2.94 9.49E)10 3.05 1.49E )09 3.05
320 5.30E)08 2.99 9.87E )08 2.82 1.18E)10 3.01 1.85E )10 3.01
DG3-HWENO5-RK3 and DG3-RK3 with no limiter, using Nequally spaced cells. t¼2. L1and L1errors of density q.
128 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
The computed density qis plotted at t¼1:3 against the exact solution. In Fig. 3 we plot the solution with
N¼200 cells by HWENO5-RK3 and WENO5-RK3, and in Fig. 4 we show the results of the DG3-
HWENO5-RK3 scheme with different TVB constants Min identifying the troubled cells, with the time
history of cells being identified as troubled cells shown in Fig. 5. We can see that both HWENO5-RK3 and
DG3-HWENO5-RK3 give equally good non-oscillatory shock transitions for this problem, and the pa-
rameter Mhas a significant effect in determining how many cells are identified as troubled cells. This in-
dicates a need for better strategy for identifying troubled cells, which we plan to investigate in the future.
x
u
-1 -0.5 0 0.5 1
0
0.5
1
x
u
-1 -0.5 0 0.5 1
0
0.5
1
Fig. 2. The Buckley–Leverett problem. t¼0:4. HWENO5-RK3 (left) and DG3-HWENO5-RK3 (right) with N¼80 cells. Solid line:
exact solution; squares: computed solution.
0 0.5 1 1.5 2
x
-0.5
0
0.5
1
1.5
u
x
u
0 0.5 1 1.5 2
-0.5
0
0.5
1
1.5
Fig. 1. Burgers equation. uðx;0Þ¼0:5þsinðpxÞ.t¼1:5=p. HWENO5-RK3 (left) and DG3-HWENO5-RK3 (right) with N¼80 cells.
Solid line: exact solution; squares: computed solution.
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 129
Fig. 3. The Lax problem. t¼1:3. HWENO5-RK3 (left) and WENO5-RK3 (right), N¼200 cells. Density q. Solid line: exact solution;
squares: computed solution.
x
Density
-5 -3 -1 1 3 5
0.6
1
1.4
x
Density
-5 -3 -1 1 3 5
0.6
1
1.4
x
Density
-5 -3 -1 1 3 5
0.6
1
1.4
Fig. 4. Lax problem by DG3-HWENO5-RK3 with 200 cells, with the TVB constant M¼0:01 (left), M¼10:0 (middle) and M¼50
(right). t¼1:3. Density q. Squares are the computed solution and solid lines are the exact solution.
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fig. 5. Lax problem by DG3-HWENO5-RK3 with 200 cells, with the TVB constant M¼0:01 (left), M¼10:0 (middle) and M¼50
(right). t¼1:3. Time history of troubled cells. Squares are the troubled cells where the HWENO limiters are used.
130 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
Example 4.6. The previous examples contain only shocks and simple smooth region solutions (almost
piecewise linear), for which shock resolution is the main concern and usually a good second-order non-
oscillatory scheme would give satisfactory results. There is little advantage in using higher order schemes
for such cases. We have been using them in the numerical experiments mainly to demonstrate the non-
oscillatory properties of the high order schemes. A higher order scheme would show its advantage when the
solution contains both shocks and complex smooth region structures. A typical example for this is the
problem of shock interaction with entropy waves [28]. We solve the Euler equations (4.2) with a moving
Mach ¼3 shock interacting with sine waves in density, i.e. initially
ðq;v;pÞ¼ð3:857143;2:629369;10:333333Þfor x<4;
ðq;v;pÞ¼ð1þesin 5x;0;1Þfor xP4:
Here we take e¼0:2. The computed density qis plotted at t¼1:8 against the reference solution, which is a
converged solution computed by the fifth-order finite difference WENO scheme [17] with 2000 grid points.
In Fig. 6 we show the results of the HWENO5-RK3 and WENO5-RK3 schemes with N¼300 cells, and
in Figs. 7 and 8 we show the results of the DG3-HWENO5-RK3 scheme with N¼200 cells and the time
history of trouble cells where HWENO limiters are used. We can see that the N¼200 results for DG3-
HWENO5-RK3 with a higher value of Mare comparable with the HWENO5-RK3 or WENO5-RK3
results with N¼300 cells.
Example 4.7. We consider the interaction of blast waves of Euler equation (4.2) with the initial condition
ðq;v;pÞ¼ð1;0;1000Þfor 0 6x<0:1;
ðq;v;pÞ¼ð1;0;0:01Þfor 0:16x<0:9;
ðq;v;pÞ¼ð1;0;100Þfor 0:96x:
Fig. 6. The shock density wave interaction problem. t¼1:8. HWENO5-RK3 (left) and WENO5-RK3 (right) with N¼300 cells.
Density q. Solid line: ‘‘Exact solution’’; squares: computed solution.
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 131
x
Density
-5 -3 -1 1 3 5
0
1
2
3
4
5
x
Density
-5 -3 -1 1 3 5
0
1
2
3
4
5
x
Density
-5 -3 -1 1 3 5
0
1
2
3
4
5
Fig. 7. The shock density wave interaction problem by DG3-HWENO5-RK3 with 200 cells, t¼1:8, with TVB constant M¼0:01
(left), M¼50:0 (middle) and M¼300 (right). Density q. Solid line: ‘‘Exact solution’’; squares: computed solution.
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x
t
-5 -3 -1 1 3 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Fig. 8. The shock density wave interaction problem by DG3-HWENO5-RK3 with 200 cells, t¼1:8, with TVB constant M¼0:01
(left), M¼50:0 (middle) and M¼300 (right). Time history of troubled cells. Squares are the troubled cells where the HWENO limiters
are used.
x
Density
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
x
Density
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
Fig. 9. The interaction of blast waves problem by HWENO5-RK3 (left) and WENO5-RK3 (right) with 400 cells, t¼0:038. Density q.
Squares are the computed solution and solid lines are the ‘‘exact’’ reference solution.
132 J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135
A reflecting boundary condition is applied to both ends. See [15,30]. The computed density qis plotted at
t¼0:038 against the reference ‘‘exact’’ solution, which is a converged solution computed by the fifth-order
finite difference WENO scheme [17] with 2000 grid points.
In Fig. 9 we show the results of the HWENO5-RK3 and WENO5-RK3 schemes with N¼400 cells, and
in Figs. 10 and 11 we show the results of the DG3-HWENO5-RK3 scheme with N¼400 cells as well as the
time history of troubled cells where HWENO limiters are used.
5. Concluding remarks
In this paper, we have constructed a new class the fifth-order WENO schemes, which we termed
HWENO (Hermite WENO) schemes, for solving nonlinear hyperbolic conservation law systems. The
construction of HWENO schemes is based on a finite volume formulation, Hermite interpolation, and
Runge–Kutta methods. The idea of reconstruction for HWENO comes from the WENO schemes. In the
HWENO schemes, both the function and its first derivative are evolved in time and used in the recon-
struction, in contrast to the regular WENO schemes where only the function value is evolved in time and
used in the reconstruction. Comparing with the regular WENO schemes, one major advantage of HWENO
schemes is their relatively compact stencil. This makes HWENO schemes more suitable for limiters in the
x
Density
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
x
Density
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
x
Density
0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
Fig. 10. The interaction of blast waves problem by DG3-HWENO5-RK3 with 400 cells, t¼0:038, with the TVB constant M¼0:01
(left), M¼50:0 (middle) and M¼300 (right). Density q. Squares are the computed solution and solid lines are the ‘‘exact’’ reference
solution.
x
t
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
x
t
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
x
t
0 0.2 0.4 0.6 0.8 1
0
0.01
0.02
0.03
0.04
Fig. 11. The interaction of blast waves problem by DG3-HWENO5-RK3 with 400 cells, t¼0:038, with the TVB constant M¼0:01
(left), M¼50:0 (middle) and M¼300 (right). Time history of troubled cells. Squares are the troubled cells where the HWENO limiters
are used.
J. Qiu, C.-W. Shu / Journal of Computational Physics 193 (2003) 115–135 133
Runge–Kutta discontinuous Galerkin methods, the application of this is also presented in this paper.
Extensive numerical experiments are performed to verify the accuracy and non-oscillatory shock resolution
of both the HWENO scheme and the RKDG method with HWENO limiters. Only the 1D case is con-
sidered in this paper. While the methodology can be generalized in principle to multi dimensions, more
work is needed to carry out the detailed design and this is left for future research.
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