COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING
Commun. Numer. Meth. Engng 2004; 20:927–937
Published online 7 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.720
Formulations and analysis of the spectral volume method
for the diusion equation
Yuzhi Sun∗; † and Z.J. Wang‡
Department of Mechanical Engineering; Michigan State University; 2555 Engineering Building;
East Lansing; MI 48824; U.S.A.
SUMMARY
The spectral volume (SV) method is a newly developed high-order nite volume method for hyperbolic
conservation laws on unstructured grids. It has been successfully demonstrated for multi-dimensional
Euler equations. We wish to extend the SV method to the Navier–Stokes equations. As a rst-step
towards achieving that goal, the SV method is extended to and tested for the diusion equation. In this
paper, we present three dierent formulations of the spectral volume method for the diusion equation.
The rst formulation yields an inconsistent and unstable scheme, while the other two formulations are
consistent, convergent and stable. A Fourier type analysis is performed for all the formulations, and the
analysis agrees well with numerical results. Copyright ? 2004 John Wiley & Sons, Ltd.
KEY WORDS:
spectral volume method; diusion equation; stability; consistency; convergence
1. INTRODUCTION
The spectral volume (SV) method [1–3] is a Godunov-type nite volume method [4], which
has been under development for several decades, and has become the state-of-the-art for the
numerical solution of hyperbolic conservation laws. For a review of the literature on the
Godunov-type method, refer to Reference [1], and the references therein. The SV method is
also related to the discontinuous Galerkin (DG) method [5]. Comparisons between the DG
and SV methods have been made recently [6, 7]. Similar to the Godunov-type method, the SV
method has two key components. One is data reconstruction, and the other is the (approximate)
Riemann solver. What distinguishes the SV method from the traditional high-order k-exact
nite volume method [8] and the weighted essentially non-oscillatory (WENO) method [9–11]
is the data reconstruction. Instead of using a (large) stencil of neighbouring cells to perform
a high-order polynomial reconstruction, a simplex unstructured grid cell—called a spectral
volume—is partitioned into a ‘structured’ set of sub-cells called control volumes (CVs), and
cell-averaged solutions on these sub-cells are then the degrees-of-freedom (DOFs). These
∗ Correspondence
to: Yuzhi Sun, Department of Mechanical Engineering, Michigan State University, 2555
Engineering Building, East Lansing, MI 48824, U.S.A.
[email protected]
[email protected]
† E-mail:
‡ E-mail:
Copyright ? 2004 John Wiley & Sons, Ltd.
Received 9 August 2003
Accepted 6 April 2004
928
Y. SUN AND Z. J. WANG
DOFs are used to reconstruct a higher-order polynomial inside the SV. If all the spectral
volumes are partitioned in a geometrically similar manner, a universal reconstruction formula
can be obtained for all simplexes. With reconstructed solutions at both sides of an interface,
the numerical ux can be computed with an approximate Riemann solver. Then the DOFs
can be updated to high-order accuracy using the usual Godunov-type nite volume method.
This high-order nite volume method has been successfully demonstrated for hyperbolic
conservation laws including non-linear systems on unstructured grids in a series of papers
[1–3]. A framework has been established to easily solve non-linear time-dependent hyperbolic
systems of conservation laws using explicit, non-linear Runge–Kutta time discretization [12]
with approximate Riemann solvers and TVB (total variation bounded) non-linear limiters [13].
Numerical tests have veried that the SV method is indeed high-order accurate, conservative
and geometrically exible.
Ultimately, we wish to extend the SV method to the Navier–Stokes equations to perform
large eddy simulation and direct numerical simulation of turbulence ow for problems with
complex geometries. To achieve this goal, we rst must nd a technique to properly discretize the second-order viscous terms. In the second-order nite volume method, the solution
gradients at an interface are computed by averaging the gradients of the neighbouring cells
sharing the face, and were shown to be adequate. For higher-order elements, special care has
to be taken in computing the solution gradients. For example, Cockburn and Shu developed
the so-called local discontinuous Galerkin method to treat the second-order viscous terms
and proved stability and convergence with error estimates [14] motivated by the successful
numerical experiments of Bassi and Rebay [15]. Baumann and Oden [16], Oden et al. [17]
introduced a dierent discontinuous Galerkin method for the discretization of the second-order
viscous terms. Riviere et al. [18] analysed three discontinuous Galerkin approximations for
solving elliptic problems in two or three dimensions. More recently, Shu [19] summarized
three dierent formulations of the discontinuous Galerkin method for the diusion equation,
and Zhang and Shu [20] performed a Fourier type analysis for these three formulations.
In this paper, we present several formulations for solving the diusion equation based on the
SV method borrowing ideas from the literature. The rst one is a nave and straightforward
implementation of the SV method for the diusion equation. The second one is obtained by
following the local discontinuous Galerkin method [14]. The third one is derived by adding
a penalty-like term into the numerical ux, which is motivated by Baumann and Oden [16],
Oden et al. [17] and Riviere et al. [18]. Numerical tests have shown that the rst formulation
is inconsistent. The second and third formulations, however, are both accurate and stable. We
perform a Fourier-type analysis on all three formulations regarding their consistency, stability
and convergence following a method presented by Zhang and Shu [20]. In the following
section, Section 2, the three SV formulations for the diusion equation are presented, together
with the numerical solutions. A Fourier-type analysis of all the formulations is given in Section
3. Conclusions from this study are summarized in Section 4.
2. THREE DIFFERENT FORMULATIONS OF THE SV METHOD
Consider the following one-dimensional diusion equation
ut = uxx ;
x ∈ [0; 2]
(1)
with periodic boundary conditions and initial condition u(x; 0) = sin(x).
Copyright ? 2004 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng 2004; 20:927–937
FORMULATIONS AND ANALYSIS OF THE SV METHOD
929
Figure 1. The numerical solutions with 40 and 320 cells versus the exact solution using the linear
reconstruction based on the Nave SV formulation for the diusion equation.
2.1. Nave SV formulation
Directly following the basic formulation described in Reference [1] for the one-dimensional
hyperbolic conservation law, we integrate (1) on control volume Ci; j , which is a sub-cell of a
spectral volume Si = [xi−1=2 ; xi+1=2 ] depicted in Figure 3, replace the ux by a numerical ux
and obtain
d ui; j (t)
1
−
(ûx |i; j+1=2 − ûx |i; j−1=2 ) = 0
hi; j
dt
(2)
Since there is no convection term in the diusion equation, the rst derivative is ‘naturally’
computed by taking a simple average of the derivatives from the two neighbouring CVs, i.e.
−
ûx |i; j+1=2 = 12 ((ux )+
i; j+1=2 + (ux )i; j+1=2 )
(3)
For time integration, we employ the third-order TVD Runge–Kutta method [14]. This formulation was used to compute a numerical solution for (1) at t = 0:7. Two dierent grids
were used in the simulation. In Figures 1 and 2 the numerical solutions with 40 and 320
SVs are compared with the exact solution using linear and quadratic reconstructions. It seems
this formulation leads to a seemingly converged, but wrong solution. Note that the numerical solutions have an O(1) error, which does not decrease with grid renement. The same
phenomenon was reported by Zhang and Shu [20].
Copyright ? 2004 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng 2004; 20:927–937
930
Y. SUN AND Z. J. WANG
Figure 2. The numerical solutions with 40 and 320 cells versus the exact solution using the quadratic
reconstruction based on the Nave SV formulation for the diusion equation.
2.2. Local SV formulation
The second formulation is obtained by mimicking the local discontinuous Galerkin method
[14]. A new variable q is introduced, which is equal to ux . Then the diusion equation becomes
the following system:
ut − qx = 0
(4)
q − ux = 0
The spectral volume method is then applied to this system directly. Integrating (4) over each
control volume, we obtain
d ui; j
1
−
(q̂|i; j+1=2 − q̂|i; j−1=2 ) = 0
dt
hi; j
qi; j −
1
(û|i; j+1=2 − û|i; j−1=2 ) = 0
hi; j
(5)
The numerical uxes are chosen following [19]:
Copyright ? 2004 John Wiley & Sons, Ltd.
û|i; j+1=2 = u|+
i; j+1=2
(6)
q̂|i; j+1=2 = q|−
i; j+1=2
(7)
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FORMULATIONS AND ANALYSIS OF THE SV METHOD
Table I. L1 and L∞ errors and orders of accuracy based on the local SV
formulation for the diusion equation.
Order of
accuracy
h
L1 error
L1 order
L∞ error
L∞ order
2
(Linear SV)
2=10
2=20
2=40
2=80
2=160
2=320
2.35e-02
6.00e-03
1.51e-03
3.78e-04
9.45e-05
2.36e-05
—
1.97
1.99
2.00
2.00
2.00
3.61e-02
9.44e-03
2.37e-03
5.94e-04
1.49e-04
3.71e-05
—
1.94
1.99
2.00
2.00
2.01
3
(Quadratic SV)
2=10
2=20
2=40
2=80
2=160
2=320
1.15e-03
1.42e-04
1.76e-05
2.20e-06
2.75e-07
3.44e-08
—
3.02
3.01
3.00
3.00
3.00
1.78e-03
2.22e-04
2.77e-05
3.46e-06
4.32e-07
5.40e-08
—
3.00
3.00
3.00
3.00
3.00
4
(Cubic SV)
2=10
2=20
2=40
2=80
2=160
2=320
6.99e-05
4.36e-06
2.72e-07
1.70e-08
1.05e-09
5.38e-11
—
4.00
4.00
4.00
4.02
4.29
1.07e-04
6.82e-06
4.27e-07
2.66e-08
1.65e-09
8.45e-11
—
3.97
4.00
4.00
4.01
4.29
i.e. we alternatively take the downwind value for u and upwind value for q (we could of
course also take the opposite pattern). Let m be the degree of the reconstruction polynomial.
Numerical solutions are computed at t = 1:0 for the three cases m = 1 (linear reconstruction),
m = 2 (quadratic reconstruction) and m = 3 (cubic reconstruction). The L1 and L∞ errors and
numerically observed orders of accuracy are presented in Table I, from which we note that a
(m + 1)th order of accuracy is achieved for a degree m polynomial reconstruction.
2.3. Penalty SV formulation
In order to remedy the rst formulation, Baumann and Oden [16], also Oden et al. [17],
and Riviere et al. [18] introduced a penalty term to the numerical ux. However if the
formulation of Baumann and Oden [16] is used directly in the SV method, the penalty term
vanishes because the weighting function is piece-wise constant in the SV method. Therefore
the Baumann and Oden formulation for the SV method is identical to the rst formulation.
Instead, a penalty-like term in the following form is added to the numerical ux for the SV
method at the interface. The SV scheme then becomes
d ui; j (t)
1
−
(ûx |i; j+1=2 − ûx |i; j−1=2 ) = 0
dt
hi; j
−
ûx |i; j+1=2 = 12 ((ux )+
i; j+1=2 + (ux )i; j+1=2 ) +
Copyright ? 2004 John Wiley & Sons, Ltd.
(u|+
− u|−
i; j+1=2 )
hi; j i; j+1=2
(8)
(9)
Commun. Numer. Meth. Engng 2004; 20:927–937
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Y. SUN AND Z. J. WANG
Table II. L1 and L∞ errors and orders of accuracy based on the penalty SV
formulation for the diusion equation.
Order of
accuracy
h
L1 error
L1 order
L∞ error
L∞ order
2
(Linear SV)
2=10
2=20
2=40
2=80
2=160
2=320
6.05e-03
1.51e-03
3.78e-04
9.46e-05
2.36e-05
5.91e-06
—
2.00
2.00
2.00
2.00
2.00
9.35e-03
2.34e-03
5.92e-04
1.48e-04
3.71e-05
9.28e-06
—
2.00
1.98
2.00
2.00
2.00
3
(Quadratic SV)
2=10
2=20
2=40
2=80
2=160
2=320
2.77e-03
6.77e-04
1.68e-04
4.20e-05
1.05e-05
2.63e-06
—
2.03
2.01
2.00
2.00
2.00
4.28e-03
1.05e-03
2.63e-04
6.60e-05
1.65e-05
4.13e-06
—
2.03
2.00
1.99
2.00
2.00
4
(Cubic SV)
2=10
2=20
2=40
2=80
2=160
2=320
6.47e-05
3.99e-06
2.48e-07
1.55e-08
9.70e-10
6.40e-11
—
4.02
4.01
4.00
4.00
3.92
1.00e-04
6.16e-06
3.88e-07
2.43e-08
1.52e-09
1.01e-10
—
4.02
3.99
4.00
4.00
3.91
where is a constant. A Fourier analysis is performed for this formulation in the case of
m = 1, and it is found that must be one to preserve second-order accuracy. Furthermore,
numerical simulations have showed that this formulation can achieve second-order accuracy
for linear and quadratic reconstructions, and fourth-order accuracy for cubic reconstructions.
Table II shows the L1 and L∞ errors and the numerically observed orders of accuracy at
t = 1:0.
3. ANALYSIS OF THE THREE FORMULATIONS
In this analysis, we follow a technique described by Zhang and Shu [20], and focus on the
linear reconstruction only. In this case, a SV is partitioned into two equal CVs, as shown
in Figure 3, and CV-averaged mean solutions are uj; 1 and uj; 2 . For simplicity of analysis
we assume that the mesh is uniform. Hence, under linear reconstruction all the three SV
formulations can be cast in the following form
uj−1; 1
uj; 1
uj+1; 1
d uj; 1
=A
+B
+C
(10)
dt uj; 2
uj−1; 2
uj; 2
uj+1; 2
where A; B and C are constant matrices. We seek general solutions of the following form:
u(x; t) = ûk (t)eikx
Copyright ? 2004 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng 2004; 20:927–937
FORMULATIONS AND ANALYSIS OF THE SV METHOD
x j ,1
x
933
x j,2
x
j ,1 / 2
x
j ,3 / 2
j ,5 / 2
Figure 3. Linear spectral volume.
where k is the index of modes (k = 1; 2; : : :) representing the wave number. Obviously, the
2
analytical solution for (1) is u(x; t) = eikx−k t . By a simple derivation, the solutions we are
looking for can be expressed as
uj; 1 (t)
ûk; 1 (t)
=
eikxj; 3=2
(11)
uj; 2 (t)
ûk; 2 (t)
Note that this analysis depends on the assumption of uniform mesh and periodic boundary
conditions. Substituting (11) into (10), we obtain the following evolution equation:
ûk; 1 (t)
ûk; 2 (t)
= G(k; h)
ûk; 1 (t)
ûk; 2 (t)
(12)
where the amplication matrix is given by
G(k; h) = e−ikh A + B + eikh C
(13)
Let 1 and 2 be the two eigenvalues of the amplication matrix G(k; h), V1 and V2 be the
corresponding eigenvectors of G(k; h), the general solution of (12) can be expressed as
ûk; 1 (t)
= e1 t V1 + e2 t V2
(14)
ûk; 2 (t)
Note that the values of and can be determined from the initial condition.
By studying the properties of this general solution at the lowest mode (k = 1), we can
obtain consistency and convergence results; by investigating the boundedness of the general
solution at the high modes (large k), we can establish the stability of the formulations.
3.1. The nave SV formulation
For the nave SV formulation, the corresponding coecient matrices A, B, and C are






2
2
2
2
0
0
−
2
− 2

h2 


; C = 
h ; B=  h
A =  h2
 2
 2
2 
2 
− 2
0
0
− 2
h
h2
h2
h
Copyright ? 2004 John Wiley & Sons, Ltd.
(15)
Commun. Numer. Meth. Engng 2004; 20:927–937
934
Y. SUN AND Z. J. WANG
and the amplication matrix G(k; h) is given by


2
2 −ikh
2
2 −ikh
e
−
−
e
+
 h2
h2
h2
h2 

G(k; h) = 
 2

2
2
2
ikh
− 2 eikh + 2
e
−
h
h
h2
h2
The eigenvalues and eigenvectors of the amplication matrix G(k; h) are
4
(1 − cos(kh)); 2 = 0
h2
−ikh 1
e
; V2 =
V1 =
1
1
1 = −
(16)
(17)
(18)
Note that G(k; h) has a zero eigenvalue, which may cause a weak instability for this semidiscrete system.
We rst study the lowest mode, i.e. k = 1. From the initial condition, the coecients and
can be computed as
=
4(1 − cos(h=2))
;
ih(e−ih − 1)
= −
2(1 − 2e−ih=2 + e−ih )
ih(e−ih − 1)
(19)
We therefore have the explicit solution of the SV scheme (10) represented by (15). For
example
uj; 1 (t) = (e1 t e−ih + e2 t )eixj; 3=2
(20)
Applying a Taylor expansion assuming small h, we obtain the imaginary part of uj; 1 (t) to be
1 + e−2t
sin(xj; 1 ) + O(h)
Im{uj; 1 (t)} =
2
where xj; 1 = 12 (xj; 1=2 + xj; 3=2 ). The solution is about 0:6233 sin(xj; 1 ) at t = 0:7, which agrees
very well with the numerical solution shown in Figure 1. We also clearly see that the scheme
is not consistent, i.e. the numerical solution does not converge to the solution of the PDE
(which equals to sin(x)e−t ).
Next we study the stability of the nave SV formulation by considering the high modes
(large k). When cos(kh) = 1, the amplication matrix G(k; h) = 0. Therefore the solution to
(10) remains to be the initial solution. When cos(kh) = 1, we can obtain explicitly the solution
of (12) as
ûk; 1 (t)
ûk; 1 (0)
G(k; h)t
=e
(21)
ûk; 2 (t)
ûk; 2 (0)
with
e
Copyright ? 2004 John Wiley & Sons, Ltd.
G(k; h)t
=R
e 1 t
0
0
1
R−1
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FORMULATIONS AND ANALYSIS OF THE SV METHOD
935
where, R is a matrix composed by eigenvectors as its columns. The L2 norm of eG(k; h)t
can be computed explicitly, which reveals the stability property. The two eigenvalues of the
T ) are found to be
symmetric matrix (eG(k; h)t )H (eG(k; h)t ) (where AH = (A)
1 = {(1 − )2 + − (1 − ) (1 − )2 + 2}=
2 = {(1 − )2 + + (1 − ) (1 − )2 + 2}=
If we take = h2 =t, we can easily get 1 = O(1=h2 ) and also
with = e1 t , = 1 − cos(kh). 2 = O(1=h2 ), then eG(k; h)t = max(|1 |; |2 |) = O(1=h), which is unbounded when h → 0.
Therefore system (12) for the nave SV scheme is unstable.
3.2. The local SV formulation
Based on the local SV formulation, we obtain the corresponding coecient matrices A; B,
and C are






12
4
3
1
1
5
−
−
 h2
 2
h2 
h2 


; C =  h

(22)
A =  h2 h2  ; B = 
 2


6
6
2 
0
0
− 2
− 2
h2
h
h2
h
The amplication matrix, its eigenvalues and eigenvectors are

3
1 −ikh
12
+ 2 eikh − 2
 h2 e
h
h
G(k; h) = 

2
6 ikh
e + 2
h2
h
1 = −

16
;
h2
2
(1 − cos(kh))
h2



1 −ikh
(e
+ 1)


 ; V2 =  2
1
2 = −
−5 + eikh
 1 + 3eikh
V1 = 
1

4
5 −ikh
1 ikh
e
−
e
+
h2
h2
h2 


2 ikh
6
− 2e − 2
h
h
(23)
(24)
(25)
Clearly both eigenvalues are real and negative. To study the accuracy and consistency, we
again examine the lowest mode k = 1. The coecients and can be found by applying
the initial condition. We thus have the following explicit solution for scheme (10) specied
by (22):
−5 + eih
2 t 1 −ih
(e
+
e
+
1)
eixj; 3=2
(26)
uj; 1 (t) = e1 t
1 + 3eih
2
Applying a Taylor expansion assuming small h, the imaginary part of uj; 1 (t) can be expressed
to be
Im{uj; 1 (t)} = sin(xj; 1 )e−t + O(h2 )
Clearly, the numerical solution converges to the exact solution with second-order accuracy.
Copyright ? 2004 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng 2004; 20:927–937
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Y. SUN AND Z. J. WANG
Also, we note R as the matrix composed by the eigenvectors given by (25). By detail
calculations, the L2 norms of R and R−1 , i.e. R and R−1 , are also uniformly bounded and
the stability of scheme (10), based on the local SV formulation, is established.
3.3. The penalty SV formulation
Finally, we turn to the analysis of the third formulation with being 1. Thus, we have the
following matrices:


A=
0
0


4
h2 
;
0

8
− 2

h

G(k; h) = 
4
4 ikh
e
h2
+

4
h2 
;
8 
− 2
h2
h

4
4 −ikh
e
+
h2
h2 


8
− 2
h
h2

8
 − h2
B= 
 4
0
C=  4
h2
0


(27)
0
(28)
The two eigenvalues of G(k; h) are
8
8
(1 + cos(kh=2)); 2 = − 2 (1 − cos(kh=2))
2
h
h
Clearly both eigenvalues are real and negative. The corresponding eigenvectors are
−ikh=2 −ikh=2 −e
e
V1 =
; V2 =
1
1
1 = −
(29)
(30)
The coecients and are in (14) are
2(eih=2 − 1)
ih
We thus have the explicit solutions of scheme (10) indicated by (27). For example
= 0;
=
uj; 1 (t) = (e1 t (−e−ih=2 ) + e2 t e−ih=2 )eixj; 3=2
(31)
(32)
Using a Taylor expansion, we obtain the imaginary part of uj; 1 (t) to be
Im{uj; 1 (t)} = sin(xj; 1 )e−t + O(h2 )
Clearly, the scheme is consistent and second-order accurate. Note that we may take ¿O(h)
for consistency, but we only have Im{uj; 1 (t)} = sin(xj; 1 )e−t + O(h) if = 1, which is why
we suggest = 1.
The matrix composed of the eigenvectors (30) of G(k; h) is R. It can be calculated that
√
√
R = 2 and R−1 = 2=2
It is clear that both R and R−1 are uniformly bounded with respect to kh. Thus the
stability of scheme (10), subjected to the penalty SV formulation, is established.
Copyright ? 2004 John Wiley & Sons, Ltd.
Commun. Numer. Meth. Engng 2004; 20:927–937
FORMULATIONS AND ANALYSIS OF THE SV METHOD
937
4. CONCLUSIONS
Three dierent formulations of the SV method are presented for the diusion equation. Numerical tests and analysis are performed for these formulations. We have found that both
the local SV and penalty-like SV formulations are stable and consistent while the nave SV
formulation is neither consistent nor stable. Numerical results agree well with the analysis.
It appears that the local SV formulation achieves (m + 1)th order accuracy with a degree m
polynomial reconstruction, while the penalty SV formulation achieves (m+1)th order accuracy
if m is odd, but mth order accuracy if m is even.
REFERENCES
1. Wang ZJ. Spectral (nite) volume method for conservation laws on unstructured grids: basic formulation. Journal
of Computational Physics 2002; 178:210–251.
2. Wang ZJ, Liu Y. Spectral (nite) volume method for conservation laws on unstructured grids III:
one-dimensional systems and partition optimization. Journal of Scientic Computing 2004; 20:137–157.
3. Wang ZJ, Zhang L, Liu Y. Spectral (nite) volume method for conservation laws on unstructured grids IV:
extension to two-dimensional Euler equations. Journal of Computational Physics 2004; 194:716–741.
4. Godunov SK. A nite-dierence method for the numerical computation of discontinuous solutions of the
equations of uid dynamics. Mathematics of the USSR-Sbornik 1959; 47:271.
5. Cockburn B, Lin S-Y, Shu C-W. TVD Runge–Kutta local projection discontinuous Galerkin nite element
method for conservation laws III: one-dimensional systems. Journal of Computational Physics 1989; 84:
90–113.
6. Zhang M, Shu C-W. An analysis of and a comparison between the discontinuous Galerkin and spectral nite
volume methods. Computers and Fluids, to appear.
7. Sun Y, Wang ZJ. Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system
conservation laws on unstructured grids. International Journal for Numerical Methods in Fluids 2004; 45:
819–838.
8. Barth TJ, Frederickson PO. High-order solution of the Euler equations on unstructured grids using quadratic
reconstruction. AIAA Paper No. 90-0013, 1990.
9. Jiang G, Shu C-W. Ecient implementation of weighted ENO. Journal of Computational Physics 1996; 126:
202.
10. Liu XD, Osher S, Chan T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics
1994; 115:200–212.
11. Shu C-W. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic
conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Cockburn B,
Johnson C, Shu C-W, Tadmor E, Quarteroni A (eds). Lecture Notes in Mathematics, vol. 1697. Springer: Berlin,
1998; 325–432.
12. Shu C-W, Osher S. Ecient implementation of essentially non-oscillatory shock capturing schemes. Journal of
Computational Physics 1988; 77:439–471.
13. Shu C-W. TVB uniformly high-order schemes for conservation laws. Mathematics of Computation 1987;
49:105–121.
14. Cockburn B, Shu C-W. The local discontinuous Galerkin method for time-dependent convection diusion system.
SIAM Journal on Numerical Analysis 1998; 35:2440–2463.
15. Bassi F, Rebay S. A high-order accurate discontinuous nite element method for the numerical solution of the
compressible Navier–Stokes equations. Journal of Computational Physics 1997; 131:267–279.
16. Baumann CE, Oden JT. A discontinuous hp nite element method for convection-diusion problems. Computer
Methods in Applied Mechanics and Engineering 1999; 175:311–341.
17. Oden JT, Babuska I, Baumann CE. A discontinuous hp nite element method for diusion problems. Journal
of Computational Physics 1998; 146:491–519.
18. Riviere B, Wheeler M, Girault V. A priori error estimates for nite element methods based on discontinuous
approximation spaces for elliptic problems. SIAM Journal on Numerical Analysis 2001; 39:902–931.
19. Shu C-W. Dierent formulations of the discontinuous Galerkin method for the viscous terms. In Advances in
Scientic Computing, Shi Z-C, Mu M, Xue W, Zou J (eds). Science Press: Beijing, 2001; 144–145.
20. Zhang M, Shu C-W. An analysis of three dierent formulations of the discontinuous Galerkin method for
diusion equations. Mathematical Models and Methods in Applied Sciences 2003; 13:395–413.
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Commun. Numer. Meth. Engng 2004; 20:927–937