J Sci Comput (2013) 54:663–683
DOI 10.1007/s10915-012-9637-0
A New Nonsymmetric Discontinuous Galerkin Method
for Time Dependent Convection Diffusion Equations
Jue Yan
Received: 31 January 2012 / Revised: 12 July 2012 / Accepted: 21 August 2012 /
Published online: 5 September 2012
© Springer Science+Business Media, LLC 2012
Abstract We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function
derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general
nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with
piecewise P k polynomial approximations. Numerical examples under one-dimensional and
two-dimensional settings are carried out. Optimal (k + 1)th order of accuracy with P k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the
Baumann-Oden method and the NIPG method, the optimal convergence is recovered for
even order P k polynomial approximations.
Keywords Discontinuous Galerkin method · Convection diffusion equation · Stability ·
Convergence
1 Introduction
In this paper, we discuss discontinuous Galerkin finite element method for solving convection diffusion equations of the form
(1.1)
Ut + ∇ · F (U ) − ∇ · A(U )∇U = 0, (x, t) ∈ Ω × (0, T ),
where Ω ⊂ Rd , the matrix A(U ) = (aij (U )) is symmetric and positive definite. U is an unknown function of (x, t) with x ∈ Ω. A new methodology is proposed to handle the diffusion
term.
Dedicated to Stanley Osher’s 70’s birthday.
The research of this author is supported by NSF grant DMS-0915247.
J. Yan ()
Department of Mathematics, Iowa State University, Ames, IA 50011, USA
e-mail: jyan@iastate.edu
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J Sci Comput (2013) 54:663–683
The discontinuous Galerkin (DG) method is a finite element method using piecewise
discontinuous functions as the approximate numerical solution and the test functions. The
combination of having discontinuous functions across the computational cells as approximations with a localized data structure makes these methods extremely flexible. As a result,
DG methods have found rapid applications in diverse areas. The application of DG methods to hyperbolic problems has been quite successful since it was originally introduced by
Reed and Hill [21] in 1973 for neutron transport equations. A major development of the DG
methods for nonlinear hyperbolic conservation laws was carried out by Cockburn, Shu, and
collaborators. We refer to [11–13] for reviews and further references.
However, the application of the DG method to diffusion problems has been a challenging
task because of the subtle difficulty in defining appropriate numerical fluxes for diffusion
terms, see e.g. [23]. There have been several DG methods suggested in literature for solving
diffusion problems. One class is the interior penalty (IP) method, which dates back to 1982
by Arnold [1] (also by Baker [3] and Wheeler [27]), the Baumann and Oden method [5, 20],
the NIPG method [22] and the IIPG method [14]. Another class is the local discontinuous
Galerkin (LDG) method introduced in [10] by Cockburn and Shu (originally proposed by
Bassi and Rebay [4] for compressible Navier-Stokes equations). We refer to [2] as the unified
analysis paper for different DG methods for diffusion. More recent works include those by
Van Leer and Nomura [26], Gassner et al. [15], and Cheng and Shu [9] and Brenner et al. [7].
Recently in [17] we developed a direct discontinuous Galerkin (DDG) method for solving diffusion equations. The scheme is based on the direct weak formulation of (1.1), thus
we name it the direct discontinuous Galerkin method. We proposed a general numerical
flux formula for the numerical solution derivative ux , which involves the average ux and
even-th order derivative jumps [∂x2m u] (m = 0, 1, . . . , [ k2 ]) to approximate the exact solution
derivative Ux at the discontinuous cell interfaces. For linear diffusion equations, an optimal
kth order error estimate in an energy norm is obtained with P k polynomial approximations.
However, numerical experiments in [17] show that the scheme accuracy is sensitive to the
coefficients in the numerical flux formula. That is, for higher order P k (k ≥ 4) polynomial
approximations it is difficult to identify suitable coefficients in the numerical flux formula to
obtain optimal (k + 1)th order of accuracy. Then in [18], we introduce extra interface correction terms in the scheme formulation and a refined version of the DDG method is obtained.
A simpler numerical flux formula is used in [18] and numerically optimal (k + 1)th order of
accuracy under L2 norm is achieved for all P k polynomials. The refined DDG method [18]
is not sensitive to the coefficients in the numerical flux formula. Numerical tests show a
large class of admissible numerical fluxes can lead to optimal order of convergence.
In this paper we introduce a notion of test function derivative numerical flux term vx ,
and add interface terms involving vx in the original DDG scheme formulation. The new DG
method thus obtained has a nonsymmetric structure. The test function numerical flux term
vx is introduced to balance the solution derivative numerical flux term ux and shares the
same format as ux . Notice we need an extra admissibility condition to guarantee the scheme
stability in [17] and [18] . For general nonlinear diffusion equations, the admissibility condition is hard to analyze. For the new nonsymmetric DG method, on the other hand, the
nonlinear stability for general nonlinear diffusion problems can be obtained without extra
admissibility condition. Optimal kth order error estimate under an energy norm is obtained
for linear diffusion problems with piecewise P k polynomial approximations.
The proposed DG method is closely related to the Baumann-Oden method [5] and the
NIPG method [22]. The essential difference is that the new nonsymmetric DG scheme has
an extra second derivative jump term included in the scheme formulation. We know both
the Baumann-Oden scheme and the NIPG scheme can obtain suboptimal kth order of accuracy with even order P k order polynomial approximations. For the new nonsymmetric DG
J Sci Comput (2013) 54:663–683
665
method, numerically an optimal (k + 1)th order of accuracy is obtained for any order P k
polynomial approximations. In a word, the optimal convergence is recovered when comparing with the Baumann-Oden method and the NIPG method.
In this paper we use uppercase capital letters to represent the exact solution and lowercase
letters to represent the DG numerical solution. The rest of the paper is organized as follows.
In Sect. 2, we describe the scheme formulation under one-dimensional setting, and present
the stability and energy norm error estimate results. Extension to two-dimensional diffusion
problems are presented in Sect. 3. Finally numerical examples are shown in Sect. 4.
2 One-Dimensional Diffusion Equations
In this section, we present the new nonsymmetric discontinuous Galerkin method for onedimensional diffusion equation,
Ut − a(U )Ux x = 0,
for (x, t) ∈ (c, d) × (0, T )
(2.1)
augmented with initial condition U (x, 0) = U0 (x) and periodic boundary condition. Here
a(U ) ≥ 0. Note that it is for simplicity of presentation to consider periodic boundary conditions. The scheme can be easily applied to any well-posed boundary conditions. Extension
to one-dimensional convection diffusion equations will be presented in Sect. 2.3. First we partition the domain Ω = (c, d) into computational cells (c, d) = N
j =1 Ij ,
where Ij = [xj − 1 , xj + 1 ], j = 1, . . . , N with x 1 as the left end and xN+ 1 as the right end
2
2
2
2
of the interval. The center of cell Ij is denoted by xj = 12 (xj − 1 + xj + 1 ) and the size of the
2
2
cell denoted by xj = xj + 1 − xj − 1 . Let’s take x = maxj xj . The numerical solution u
2
2
and the test function v are both piecewise polynomials of degree k. For any time t ∈ [0, T ],
u ∈ Vx , where
Vx := v ∈ L2 (c, d) : v|Ij ∈ P k (Ij ), j = 1, . . . , N ,
and P k (Ij ) denotes the space of polynomials in Ij with degree at most k.
2.1 Scheme Formulation and Nonlinear Stability
Let’s use the notation b(s) = a(s)ds. To formulate the original direct discontinuous
Galerkin (DDG) method [17], we multiply the diffusion equation (2.1) by an arbitrary test
function v(x) ∈ Vx , integrate over the cell Ij , have the integration by parts, and formally
we obtain,
1
x v j + 21 +
ut vdx − b(u)
a(u)ux vx dx = 0,
(2.2)
j−
2
Ij
Ij
where
1
−
x v j + 21 := b(u)
x
x
b(u)
−
b(u)
v+ .
1v
1
j
+
j− 1 j− 1
j+
j−
2
2
2
2
2
Note here and below we adopt the following notations:
u± = u(x ± 0, t),
[u] = u+ − u− ,
u=
u+ + u−
.
2
(2.3)
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Motivated by the solution derivative trace formula of the heat equation with discontinuous
x with the following
initial data, in [17] we introduced a numerical flux formula for b(u)
form,
x = β0 [b(u)] + b(u)x + β1 x b(u)xx + β2 (x)3 b(u)xxxx + · · · .
b(u)
x
(2.4)
x which approximates a(u)ux = b(u)x at the discontinuous cell
The numerical flux b(u)
interface xj ±1/2 thus involves the average b(u)x and the even order derivatives jumps of
b(u). The coefficients β0 , β1 , . . . are chosen to ensure the stability and the convergence of
the method. This is the original direct discontinuous Galerkin method introduced in [17].
Now we observe that the test function slope vx will contribute at the interfaces whenever
x (vx for linear diffusion case),
[u] is nonzero. We thus introduce numerical flux term b(v)
x [u] in the above DDG scheme (2.2) and obtain a new nonsymmetric
add interface terms b(v)
DG scheme as follows. ∀v(x) ∈ Vx , the DG solution u(x, t) ∈ Vx of (2.1) satisfy,
⎧
1
⎨ u v dx − b(u)
x v|j + 21 + a(u)ux vx dx − (b(v)
x [u]j +1/2 + b(v)
x [u]j −1/2 ) = 0,
Ij t
Ij
j− 2
⎩ u(x, t = 0)v(x) dx =
Ij
Ij U0 (x)v(x) dx.
(2.5)
The numerical fluxes are defined as
x = β0u [b(u)] + b(u)x + β1 x[b(u)xx ],
b(u)
x
x = β0v [b(v)] + b(v)x + β1 x[b(v)xx ].
b(v)
x
(2.6)
x +x
Here x = j 2 j +1 when the numerical flux is evaluated at the cell interface xj +1/2 .
Notice we drop the higher order terms in (2.4) and take a simpler numerical flux formula for
x . To balance the scheme, b(v)
x is taken in the same format as b(u)
x . As a discontinuous
b(u)
Galerkin method, we usually take the test function v(x) to be nonzero only inside the cell Ij
that are evaluated outside of Ij will
to resolve the solution u(x) in cell Ij , thus terms in b(v)
be zero. In a word, the b(v)x terms in (2.5) can be written out explicitly as the following
−
x )j +1/2 = −β0v b(v ) + 1 b(v − )x − β1 xb(v − )xx ,
(b(v)
x
2
+
x )j −1/2 = β0v b(v ) + 1 b(v + )x + β1 xb(v + )xx .
(b(v)
x
2
x [u] essentially equals to the
The v ± is as defined in (2.3). Now we see the first term of b(v)
first term of b(u)x v at the cell interfaces, thus only the difference β0 = β0u − β0v matters
in the scheme formulation (2.5). This completes the definition of the new nonsymmetric
discontinuous Galerkin method for diffusion equation (2.1). Next we discuss the stability
issues of the nonsymmetric DG method.
Theorem 2.1 (Energy stability) Consider the nonsymmetric DG scheme (2.5)–(2.6) for general nonlinear diffusion equation (2.1), we have
J Sci Comput (2013) 54:663–683
1
2
T
u2 (x, T )dx +
0
Ω
≤
1
2
667
N j =1
N
T
a(u)u2x (x, t)dxdt + β0
0
Ij
j =1
a(uξj )
[u]2
dt
x
U02 (x)dx,
(2.7)
Ω
+
with β0 = β0u − β0v and uξj ∈ [u−
j +1/2 , uj +1/2 ].
Proof Sum the DG scheme (2.5) over all cells Ij , j = 1, . . . , N , we obtain the primal weak
formulation of (2.1) as the following
ut vdx +
Ω
N a(u)ux vx dx +
Ij
j =1
N
x [v]j +1/2 −
b(u)
j =1
N
x [u]j +1/2 = 0.
b(v)
(2.8)
j =1
Take v = u in the above primal weak formulation, and with the numerical fluxes given in
(2.6) we have,
1 d
2 dt
u2 dx +
Ω
N j =1
Ij
a(u)u2x dx + β0
N
[b(u)]
j =0
x
[u]j +1/2 = 0,
(2.9)
+
where β0 = β0u − β0v . Notice there exists uξj between u−
j +1/2 and uj +1/2 such that,
b(u) j +1/2 = b u+ − b u− = a(uξj )[u].
Now integrate the equality (2.9) over time [0, T ], and consider the initial condition setup as
in (2.5), we directly obtain the nonlinear stability result.
Remark 2.1 Checking out the above stability theorem, we see our nonsymmetric DG method
shares the same stability result (2.7) with the Baumann-Oden method (β0 = 0) and the NIPG
method (β0 > 0). The essential difference is that our nonsymmetric DG scheme has the
second derivative jump terms included in the numerical fluxes (2.6). On the other hand,
numerically we obtain optimal (k + 1)th order of convergence with any P k polynomial approximations. It is well known that both the Baumann-Oden scheme and the NIPG method
can only obtain sub-optimal kth order with even order polynomial approximations. Notice
that the second order derivative jump terms only contribute with higher order (k ≥ 2) polynomial approximations. Our scheme degenerates to the Baumann-Oden scheme or the NIPG
scheme with piecewise constant or piecewise linear approximations, at least for the linear
problems. Thus in the following numerical section, we focus on high order polynomial approximations.
1
) as in [17] for the
Remark 2.2 We can choose the favorite coefficient pair (β0 , β1 ) = (2, 12
numerical fluxes (2.6). Numerical experiments show we need smaller β1 coefficient for high
order P k (k ≥ 4) approximations.
Remark 2.3 We should point out the assumption of periodic boundary condition is for simplicity of presentation. Notice the scheme (2.5) is based on the direct weak formulation of
the diffusion equations, it’s natural to apply Neumann type of boundary conditions to the
numerical flux ux term at the domain boundary.
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2.2 Energy Norm Error Estimate for 1-D Linear Diffusion Equation
In this section, we discuss the error estimate of the nonsymmetric DG solution for the linear
diffusion equation with a(U ) = 1 of (2.1). We adopt e := u − U to denote the error between
the numerical solution u and the exact solution U , and the energy norm
v(·, t) :=
v dx +
2
t
N 0 j =1
Ω
vx2 dxdτ
Ij
1/2
t
N
[v]2
dτ
+ β0
0 j =1 x
(2.10)
with β0 = β0u − β0v > 0 of (2.6). The form of this energy norm is inspired by the stability
estimate (2.7). Before carrying on the error estimate, we first list the following lemmas as
the approximation properties of the finite element space Vkx . We refer to the finite element
textbook by Brenner and Scott [6] for the reference of the first two lemmas.
Lemma 2.1 (Approximation property) Let K ⊂ Rn be any regular element in the sense
that ρx ≤ diam(K) ≤ x for some constant ρ. Let U ∈ W k+1,p (Ω) and P(U ) be the L2
projection of U in Vkx . Then we have the following approximation property,
U − P(U ) m,q ≤ ck (x)n/q−n/p |U |W k+1,p (K) (x)k+1−m .
(2.11)
W
(K)
Here p, q ∈ [1, ∞], m ≥ 0 and k ≥ 0 are integers, and the constant ck solely depends on k.
Lemma 2.2 (Inverse inequality) Given the finite dimensional piecewise polynomial space
Vkx , and 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and 0 ≤ m ≤ l, and a regular element K ⊂ Rn , there
exists C independent of x such that for all v ∈ Vkx , we have
vW m,q (K) ≤ Cx l−m+n/q−n/p vW l,p (K) .
(2.12)
Lemma 2.3 Let U be the smooth exact solution. Then we have the projection error
U − P(U ) (·, T ) ≤ C ∂ k+1 U (x)k .
x
Proof Apply Lemma 2.1 with p = q = 2, we immediately obtain the estimate,
N
P(U ) − U 2
0,Ij
j =1
T
+
0
N
P(U ) − U 2 dt
1,I
j
j =1
≤ Ck (x)2k+2 |U |2k+1,Ω + (x)2k
dt
.
k+1,Ω
T
U (·, t)2
0
Take p = 2, q = ∞ and m = 0 in Lemma 2.1, we have,
0
T
N
[P(U ) − U ]2
j =1
x
T
U (·, t)2
dt ≤ (x)2k
0
k+1,Ω
dt.
Combine the above two results, we have the estimate of |||(U − P(U ))(·, T )|||2 .
From the stability analysis and the approximation properties of the finite element space
Vx , we obtain the following error estimate.
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669
Theorem 2.2 (Error estimate) Let e := u − U be the error between the exact solution U
and the numerical solution u of the nonsymmetric DG method (2.5)–(2.6), then the energy
norm of the error satisfies the inequality
e(·, T ) ≤ C ∂ k+1 U (·, T )(x)k ,
x
(2.13)
where C = C(k, β0 .β1 ) is a constant depending on k, the polynomial degree of the solution
space Vx , and the coefficients β0 , β1 in the numerical flux formula, but is independent of
U and x.
Proof We first rewrite the error as
e = u − P(U ) + P(U ) − U = P(e) − U − P(U ) .
(2.14)
Here P(U ) is the L2 projection of U in Vx , such that for j = 1, . . . , N
P(U )(x) − U (x) v(x)dx = 0,
∀ v(x) ∈ Vx .
Ij
Then we have
e(·, T ) ≤ P(e)(·, T ) + U − P(U ) (·, T ).
(2.15)
By the triple norm definition (2.10) and the projection properties (Lemma 2.3) we have
U − P(U ) (·, T ) ≤ C ∂ k+1 U (·, T )(x)k ,
x
thus we only need to estimate the term |||P(e)(·, T )|||. Here and below C represents a generic
constant.
Now let’s define the bilinear form B(w, v) as
T
T
wt v dxdt +
B(w, v) =
0
0
Ω
N j =1
wx vx dxdt + Θ(T , w, v)
(2.16)
Ij
with
T
Θ(T , w, v) =
0
N
j =1
w
x [v] j +1/2 dt −
0
T
N
j =1
vx [w] j +1/2 dt.
(2.17)
The numerical fluxes w
x and vx are as defined in (2.6). By the scheme defition, we have
B(u, v) = 0 and B(U, v) = 0, ∀v ∈ Vx . In a word, we have B(e, v) = B(u − U, v) = 0, or
its equivalence B(P(e), v) = B(U − P(U ), v) by (2.14).
Taking v = u − P(U ) = P(e), we have
B P(e), P(e) = B U − P(U ), P(e) .
(2.18)
From (2.5) we have P(e)(·, 0) = 0, thus we obtain the left hand side of (2.18) as,
2 1 2
B P(e), P(e) = P(e)(·, T ) − P(e)(·, T ) .
2
(2.19)
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For the right hand side of (2.18), we have,
B U − P(U ), P(e)
T
b
=
U − P(U ) t P(e)dxdt
0
a
T
+
N 0
j =1
Ij
U − P(U ) x P(e)x dxdt + Θ T , U − P(U ), P(e) ,
(2.20)
with
Θ T , U − P(U ), P(e)
T
=
N
0
j =1
U−
P(U ) x P(e) j +1/2 dt −
T
N
0
j =1
x U − P(U )
P(e)
dt.
j +1/2
(2.21)
For the first term of (2.20), with P(e) ∈ Vkx we have
b
a
U − P(U ) t P(e) dx = 0.
For the second term of (2.20),
T
0
N j =1
Ij
2
1
U − P(U ) x P(e)x dxdt ≤ C ∂xk+1 U (·, T ) x 2k +
4
T
0
N
P(e)x 2 dt,
I
j =1
j
here Cauchy inequality and projection error estimate as in Lemma 2.3 are used. For the third
term of (2.20), we first rewrite Θ(T , U − P(U ), P(e)) = I1 + I2 . With Cauchy’s inequality,
we can bound the I1 term as,
T
I1 ≤ 1
0
N
[P(e)]2j +1/2
j =1
x
dt +
x
4
1
T
0
N
j =1
2
U−
P(U ) x j +1/2 dt.
Then incorporating the projection error estimate (Lemma 2.3), we further estimate the I1
term as,
T
N
2
[P(e)]2j +1/2
I1 ≤ 1
dt + C ∂xk+1 U (·, T ) x 2k ,
x
0 j =1
x defined as
here 1 is a constant to be chosen. Notice we have the numerical flux term P(e)
x = β0v [P(e)] + P(e)x + β1 x P(e)xx .
P(e)
x
With P(e) ∈ Vkx , and the inverse inequality as in Lemma 2.2, we have
N
j =1
P(e)x
2
≤
j +1/2
N C P(e)2x dx,
x j =1 Ij
N N
2
C P(e)2x dx.
P(e)xx j +1/2 ≤
3
x
I
j
j =1
j =1
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Now use the Cauchy’s inequality twice and incorporate the above two estimates, we can
bound the I2 term in Θ (2.21) as,
T
I2 ≤ 2 β0v
0
N
[P(e)]2j +1/2
x
j =1
T
+ 3 C
0
N j =1
dt
P(e)2x dxdt +
Ij
1
1
+
4
2 4
3
T
0
N
[U − P(U )]2j +1/2
x
j =1
dt.
We can choose suitable constants 1 , 2 , 3 (
3 C = 1/4 and 1 + 2 β0v = β0 /2) and obtain
the estimate of Θ term as below,
2
1
Θ T , U − P(U ), P(e) ≤ C ∂xk+1 U (·, T ) x 2k +
4
+
β0
2
0
T
N
[P(e)x ]2j +1/2
j =1
x
T
0
N
P(e)x 2 dt
I
j =1
j
.
Finally we have the right hand side of (2.20) as
2
1 B U − P(U ), P(e) ≤ P(e)(·, T ) −
2
1
P(e)(·, T )2 + C ∂ k+1 U (·, T )2 x 2k .
x
2
With the left hand side estimate (2.19), we obtain |||P(e)(·, T )||| ≤ C|||∂xk+1 U (·, T )|||(x)k .
The needed result follows.
2.3 Extension to Convection Diffusion Equations
In this section we extend the nonsymmetric DG method to the following nonlinear convection diffusion equations,
(2.22)
Ut + f (U )x − a(U )Ux x = 0 in (c, d) × (0, T ),
subject to initial data U (x, 0) = U0 (x) and periodic boundary conditions. Again, the diffusion coefficient a(U ) ≥ 0 is assumed to be non-negative.
The nonsymmetric DG scheme is defined as follows: for j = 1, . . . , N and ∀v ∈ Vx , we
have,
1
x v j + 21
ut vdx + f
(u) − b(u)
j−
Ij
2
−
x [u]j +1/2 + b(v)
x [u]j −1/2 = 0.
f (u) − a(u)ux vx dx − b(v)
(2.23)
Ij
u
x and b(v)
x are as defined in (2.6). For
Again b(u) = 0 a(s)ds. The numerical fluxes b(u)
the convection part we can choose any entropy satisfying numerical flux, for example, the
Lax-Friedrichs flux,
1 f
(u) = f u− , u+ = f u− + f u+ − α u+ − u− ,
2
(2.24)
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J Sci Comput (2013) 54:663–683
where α = maxu∈[u− ,u+ ] |f (u)|.
Up to now, we have taken the method of lines approach and have left time variable t
continuous. For time discretization we use the explicit third order TVD Runge-Kutta method
[24, 25] to match the accuracy in space.
3 Two-Dimensional Diffusion Equations
In this section, we study the nonsymmetric DG method for two-dimensional problems. Since
the numerical flux for convection term can be treated via dimension-wise extension, we focus on the scheme formulations for nonlinear diffusion problems under the two-dimensional
setting. Now we consider the 2-D nonlinear parabolic equation,
(3.1)
Ut − ∇ · A(U )∇U = 0, (x, t) ∈ Ω × (0, T ),
subject to initial data U (x, 0) = U0 (x) and periodic boundary conditions. x = (x1 , x2 ) ∈
symmetric positive definite. Similar to
Ω ⊂ R2 . The matrix A(U ) = (aij (U )) is assumed
the one-dimensional case, we denote bij (s) = aij (s) ds, i = 1, 2, j = 1, 2.
Let Tx = {K} be a shape-regular partition of the domain Ω with elements K and denote
x = maxK diam(K). As before, define P k (K) as the space of polynomials with degree at
most k in the element K. The piecewise polynomial DG solution space is defined as,
Vkx = v ∈ L2 (Ω) : v|K ∈ P k (K), ∀K ∈ Tx .
Along the element boundary ∂K, we use v intK to denote the value of v evaluated from the
inside of element K. Correspondingly we use v extK to denote the value of v evaluated from
the outside of element K (inside the neighboring element). The average and jump of v on
edge ∂K are defined as
v=
1 extK
v
+ v intK ,
2
[v] = v extK − v intK .
3.1 Scheme Formulation for 2-D Model Equation
For sake of presentation, we first consider the case where A(U ) = I in (3.1). This gives us
the below 2-D heat equation,
Ut − U = 0,
(x, t) ∈ Ω × (0, T ).
(3.2)
As in the 1-D case, multiply the equation by test function, integrate over the computational
cell K, perform integration by parts, add extra interface terms involving test function derivative, and we have the following nonsymmetric DG scheme formulation. We seek the DG
solution u ∈ Vkx of (3.2) such that for all test functions v ∈ Vkx and on all elements K we
have,
ut v dx +
∇u · ∇v dx −
un v intK ds −
vn [u] ds = 0.
(3.3)
K
K
∂K
∂K
where the numerical flux at the cell boundary ∂K is defined as
⎧
[u]
⎨un = ∇u
· n = β0u x
+ ∂u
+ β1 x[unn ],
∂n
[v]
∂v
⎩vn = ∇v
· n = β0v + + β1 x[vnn ].
x
∂n
(3.4)
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Note, in the numerical flux definition, x is the average of the diameter of K and the diameter of its neighboring element. Here n = (n1 , n2 ) is the outward unit normal along the
element boundary ∂K and unn is the second order normal derivative. If the element boundaries are straight lines, such as the triangular meshes, the numerical flux (e.g. un ) can be
further simplified as
un = u
x1 n 1 + u
x2 n 2 ,
with
[u]
u
x1 = β0u x n1 + ux1 + β1 x[ux1 x1 n1 + ux2 x1 n2 ]
[u]
u
x2 = β0u x n2 + ux2 + β1 x[ux1 x2 n1 + ux2 x2 n2 ].
Again, the test function v is taken to be zero outside the element K, thus only one side
(inside of K) contributes to the computation of vn along the element boundary ∂K. Then,
as in the 1-D case, we obtain the following stability result.
Theorem 3.1 (Stability for the 2-D model equation) Consider the nonsymmetric DG scheme
(3.3)–(3.4) for the 2-D linear diffusion equation (3.2), we have
1
2
T
u2 (x, T ) dx +
0
Ω
K∈T
T
|∇u|2 dxdt + β0
0
K
K∈T
∂K
1
[u]2
dsdt ≤
x
2
U02 (x) dx,
Ω
(3.5)
with β0 = β0u − β0v .
This can be proved directly by summation over all K ∈ Tx of (3.3) with v = u. For the 2-D
linear model equation (3.2), we have the following energy norm error estimate. The proof is
similar to the one dimensional case given in previous section and is omitted here.
Theorem 3.2 (Error estimate in two dimensions) Consider the 2-D linear model equation
(3.2). Let e := u − U be the error between the exact solution U and the numerical solution
u of the nonsymmetric DG method (3.3)–(3.4), we have
e(·, T ) ≤ C U L∞ (0,T ;H k+1 ) + U L2 (0,T ;H k+1 ) xk ,
where C is a constant depending on k, β0 , β1 only.
3.2 Scheme Formulation for 2-D Nonlinear Diffusion Equations
We consider the fully nonlinear 2-D case as in (3.1). We seek the nonsymmetric DG solution
u ∈ Vkx of (3.1) such that ∀v ∈ Vkx and on all elements K we have,
ut v dx +
2
K
K i,j =1
−
bij (u)xj vxi dx −
2
∂K i,j =1
b
ij (v)xj ni [u] ds = 0.
2
intK
b
ds
ij (u)xj ni v
∂K i,j =1
(3.6)
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With triangular mesh, the numerical flux terms are defined as (j = 1, 2),
⎧
[bij (u)]
⎨b
ij (u)xj = β0u x nj + bij (u)xj + β1 x[bij (u)x1 xj n1 + bij (u)x2 xj n2 ]
(3.7)
[bij (v)]
⎩b
ij (v)xj = β0v x nj + bij (v)xj + β1 x[bij (v)x1 xj n1 + bij (v)x2 xj n2 ].
Similar to the 1-D case, we obtain the following nonlinear stability result for (3.1).
Theorem 3.3 (Nonlinear stability) Assume that for p ∈ R, and we have γ and γ ∗ such that
the eigenvalues of the diffusion coefficient matrix (aij (p)) lie between [γ , γ ∗ ]. Consider the
nonsymmetric DG scheme (3.6)–(3.7), the numerical solution satisfies
1
2
T
u (x, T ) dx +
2
0
Ω
≤
1
2
2
K∈T
T
aij (u)uxj uxi dxdt + γβ0
K i,j =1
U02 (x) dx,
0
K∈T
∂K
[u]2
dsdt
x
(3.8)
Ω
with β0 = β0u − β0v .
4 Numerical Examples
In this section, we provide numerical examples to illustrate the performance of the nonsymmetric DG method. One and two dimensional linear and nonlinear problems are considered.
We should specify that the 2-D numerical examples are implemented on rectangular meshes.
Example 4.1 (Linear diffusion equation (heat equation))
Ut − Uxx = 0 x ∈ [0, 2π],
(4.1)
with initial condition U (x, 0) = sin(x). We start with this simple example to test the optimal
convergence of the nonsymmetric DG method. We first consider a uniform mesh with mesh
size x = 2π
. With refined mesh, the optimal (k + 1)th order of accuracy is obtained for
N
all P k piecewise polynomial approximations, see Table 1. The error is measured under L2
and L∞ norm. We sample 100 point values per computational cell to evaluate the L∞ error.
Notice the optimal convergence is obtained with zero penalty coefficient β0 = β0u − β0v = 0,
which is quite an improvement compared to the Oden-Baumann scheme [5]. When comparing to the NIPG method [22] with positive penalty coefficient β0 = β0u − β0v > 0, similar
optimal convergence is recovered with even order polynomial approximations. In Table 2,
we show the error measured under energy norm (2.10) and optimal kth order of accuracy is
obtained.
Next, we test the convergence of the nonsymmetric DG method on a nonuniform mesh
with repeated pattern of 1.1x and 0.9x. Again, (k + 1)th order of accuracy is obtained
with P k polynomial approximations, see Table 3. Numerically we observe positive penalty
coefficient β0 will help maintaining the optimal convergence when computations are applied
on dramatic nonuniform meshes.
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675
Table 1 1-D linear diffusion equation (4.1). L2 and L∞ errors and orders at t = 0.5 on uniform mesh.
P k polynomial approximations with k = 2, 3, 4, 5, 6
k
k=2
k=3
k=4
k=5
k=6
β0
0
0
0
0
0
N = 10
N = 20
Error
Error
Order
Error
Order
Error
Order
L2
6.59E-04
7.79E-05
3.0
9.60E-06
3.0
1.20E-06
3.0
L∞
1.23E-03
1.51E-04
3.0
1.89E-05
3.0
2.36E-06
3.0
L2
1.39E-04
9.30E-06
3.9
5.92E-07
4.0
3.72E-08
4.0
L∞
2.30E-04
1.53E-05
3.9
9.79E-07
4.0
6.16E-08
4.0
L2
7.21E-07
2.14E-08
5.1
6.59E-10
5.0
2.05E-11
5.0
L∞
1.52E-06
4.60E-08
5.0
1.43E-09
5.0
4.45E-11
5.0
β1
1/12
1/12
1/40
1/40
1/80
N = 40
N = 80
N =4
N =8
L2
1.41E-05
1.59E-07
6.4
N = 12
1.29E-08
6.2
N = 16
2.22E-09
6.1
L∞
2.37E-05
3.19E-07
6.2
2.73E-08
6.1
4.81E-09
6.0
L2
1.08E-06
5.39E-09
7.6
2.74E-10
7.3
3.57E-11
7.1
L∞
1.74E-06
8.25E-09
7.7
4.15E-10
7.3
5.61E-11
7.0
Table 2 1-D linear diffusion equation (4.1). Energy norm (2.10) errors and orders at t = 0.5 on uniform
mesh. P k polynomial approximations with k = 2, 3, 4
k
β0
β1
N = 10
N = 20
Error
Error
Order
N = 40
Error
Order
N = 80
Error
Order
k=2
2
1/12
6.03E-03
1.48E-03
2.0
3.69E-04
2.0
9.22E-05
2.0
k=3
2
1/12
4.86E-04
5.69E-05
3.1
6.89E-06
3.0
8.49E-07
3.0
k=4
2
1/40
1.87E-04
1.16E-05
4.0
7.28E-07
4.0
4.55E-08
4.0
Table 3 1-D linear diffusion equation (4.1). L2 and L∞ errors and orders at t = 0.5 on nonuniform mesh.
P k polynomial approximations with k = 2, 3, 4
k
k=2
k=3
k=4
β0
4
4
10
β1
1/12
1/12
1/40
N = 10
N = 20
Error
Error
Order
N = 40
Error
Order
N = 80
Error
Order
L2
6.56E-04
8.18E-05
3.0
1.03E-05
3.0
1.35E-06
2.9
L∞
1.60E-03
1.99E-04
3.0
2.51E-05
3.0
3.18E-05
3.0
L2
2.55E-05
1.60E-06
4.0
1.00E-07
4.0
6.26E-09
4.0
L∞
6.98E-05
4.32E-06
4.0
2.71E-07
4.0
1.70E-08
4.0
L2
8.98E-07
2.57E-08
5.1
8.35E-10
4.9
3.21E-11
4.7
L∞
2.50E-06
7.40E-08
5.1
2.33E-09
5.0
7.75E-11
4.9
Example 4.2 (Fully nonlinear convection diffusion equation)
Ut + U 2 x − U 2 xx = f (x, t), x ∈ [0, 3],
(4.2)
with the exact solution constructed as U (x, t) = ex−t . Exact solution is used as the given
boundary condition and Lax-Friedrichs type numerical flux is used for the nonlinear convection term. We check out the convergence of the nonsymmetric DG scheme (2.23) and
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Table 4 1-D fully nonlinear convection diffusion equation (4.2). L2 and L∞ errors and orders at t = 1.0.
P k polynomial approximations with k = 2, 3, 4
k
β0
k=2
k=3
k=4
2
2
2
N = 10
N = 20
Error
Error
Order
Error
Order
Error
Order
L2
6.09E-04
6.48E-05
3.2
1.83E-05
3.1
7.55E-06
3.1
L∞
1.52E-03
1.96E-04
3.0
5.85E-05
3.0
2.48E-05
3.0
L2
8.08E-06
5.08E-07
4.0
1.01E-07
4.0
3.18E-08
4.0
L∞
2.71E-05
1.83E-06
3.9
3.71E-07
3.9
1.19E-07
4.0
L2
2.88E-07
8.42E-09
5.1
1.09E-09
5.0
2.61E-10
5.0
L∞
9.33E-07
2.99E-08
5.0
3.95E-09
5.0
9.24E-10
5.0
β1
1/12
1/12
1/40
N = 40
N = 80
obtain the optimal (k + 1)th order of accuracy with P k polynomial approximations, see Table 4. Also, super convergence phenomena at the cell centers {xj } is observed with even
order approximations (k = even), and (k + 2)th order of accuracy is obtained at the cell centers. we should mention that the super convergence is not observed for the interior penalty
method [1].
Example 4.3 (Nonlinear reaction diffusion equation (Fisher–Kolmogorov equation))
Ut = Uxx + U (1 − U ),
x ∈ [−20, 20].
(4.3)
√
√
− x−ct
With the wave speed c = 5/ 6, traveling wave solutions exist as U (x, t) = (1 + Ae 6 )−2
with arbitrary positive constant A > 0. We use the nonlinear reaction diffusion equation to
test the accuracy and the capability of the nonsymmetric DG method. We compute the wave
propagation at final time t = 2.0, and list the errors and orders of accuracy in Table 5.
(k + 1)th order of accuracy under both L2 and L∞ norm is obtained with P k polynomial
approximations. We also test the behavior of the Fisher equation with the following discontinuous initial condition,
1, x ≤ 0,
U (x, 0) =
0, x > 0.
Propagation of the solution at t = 0, 1, 3, 5, 7 is plotted in Fig. 1. We see the initial discontinuous profile propagates to the right with a finite wave speed, the nonsymmetric DG
solution is comparable to those in literature [8].
Example 4.4 (Nonlinear porous medium equation)
Ut − U m xx = 0,
x ∈ [−6, 6],
(4.4)
with zero boundary condition. The Barenblatt solution is given as,
U (x, t) =
⎧
1
⎨ (t + 1)− m+1
(3 −
⎩ 0,
x2
m−1
),
2m(m+1) (t+1)2/m+1
|x| ≤
|x| ≥
1
6m(m+1)
(t
m−1
+ 1) m+1 ,
6m(m+1)
(t
m−1
+ 1) m+1 .
1
(4.5)
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Table 5 1-D nonlinear reaction diffusion equation (4.3). L2 and L∞ errors and orders at t = 2.0. P k polynomial approximations with k = 2, 3, 4
k
β0
k=2
k=3
k=4
2
2
2
N = 40
N = 80
Error
Error
Order
Error
Order
Error
Order
L2
2.33E-05
2.90E-06
3.0
8.58E-07
3.0
3.62E-07
3.0
L∞
1.10E-04
1.40E-05
3.0
4.12E-06
3.0
1.73E-06
3.0
L2
1.91E-06
1.20E-07
4.0
2.36E-08
4.0
7.48E-09
4.0
L∞
9.47E-06
5.71E-07
4.1
1.09E-07
4.1
3.36E-08
4.1
L2
2.73E-08
8.17E-10
5.0
1.07E-10
5.0
2.52E-11
5.0
L∞
1.35E-07
4.49E-09
4.9
5.99E-10
5.0
1.43E-10
5.0
β1
1/12
1/12
1/40
N = 120
N = 160
Fig. 1 Fisher equation (4.3) with
discontinuous initial data.
Piecewise quadratic
approximations, mesh N = 200
For these nonlinear degenerate parabolic problems with low regularity solutions, we apply
the following modified nonsymmetric DG scheme,
Ij
1
x v j + 21 +
ut vdx − b(u)
j−
2
Ij
a(u)ux vx dx − vx b(u) j +1/2 + vx b(u) j −1/2 = 0, (4.6)
x and vx as in (2.6). Compared to (2.5), the modificawith the same numerical flux b(u)
tion is applied on the extra interface terms where we have vx [b(u)]j ±1/2 instead of terms
x [u]j ±1/2 in the scheme formulation. The modification is mainly based on numerical
b(v)
tests. We compute the compactly supported solution propagation up to t = 1.0 with quadratic
polynomial approximations, see Fig. 2 with m = 2, 3, 5, 8 in (4.4). For the coefficients pair
(β0 , β1 ) in the numerical flux, we take (β0 , β1 ) = (2, 1/12) for m = 2, 3 in (4.4). For the
stronger nonlinearity case with m = 8, we take (β0 , β1 ) = (4, 1/80).
We now consider the collision of the two-box solution with same or different heights.
The first case is for m = 5 in (4.4), with the same height two box initial profile given below,
U (x, 0) =
1, x ∈ (−3.7, −0.7) ∪ (0.7, 3.7),
0, otherwise.
(4.7)
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Fig. 2 1-D nonlinear porous medium equations (4.4) with m = 2, 3, 5, 8. Piecewise quadratic polynomial
approximations with mesh N = 80. Solid lines are exact solutions, symbol circles are the nonsymmetric DG
solutions
The evolution of the nonsymmetric DG solution at time t = 0, 0.2, 0.8, 1.5 is listed in
Fig. 3. The second case is for m = 6 in (4.4), with different height two box initial profile
given as,
⎧
⎨ 1, x ∈ (−4, −1),
U (x, 0) = 2, x ∈ (0, 3),
(4.8)
⎩
0, otherwise.
The evolution of the nonsymmetric DG solution at time t = 0, 0.01, 0.04, 0.06, 0.12, 0.8 is
listed in Fig. 4. We see in both two cases the two box propagates with finite speed, and the
solutions even out the height after long time run. The nonsymmetric DG simulations agree
well with those in literature [19].
Example 4.5 (Strongly degenerate nonlinear convection diffusion equation)
Ut + U 2 x − ν(U )Ux x = 0,
with = 0.1, and
ν(U ) =
0,
1,
x ∈ (−2, 2),
(4.9)
|U | ≤ 0.25
|U | > 0.25.
We see the equation is hyperbolic when |U (x, t)| ≤ 0.25 and is parabolic otherwise. We
solve this strongly degenerate convection diffusion problem with the following initial con-
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679
Fig. 3 Nonlinear porous medium equation with m = 5 in (4.4) and initial profile (4.7). Two box collision
with the same heights. Piecewise quadratic approximations with mesh N = 160
dition,
⎧
⎪
⎨ 1,
U (x, 0) = −1,
⎪
⎩
0,
− √12 − 0.4 < x < − √12 + 0.4,
√1
2
− 0.4 < x <
√1
2
+ 0.4,
(4.10)
otherwise.
Zero boundary condition is applied. With high order P k (k ≥ 2) polynomial approximations,
we apply the minmod type slope limiter to compress the oscillations. Piecewise quadratic
approximations of the nonsymmetric DG solution at time t = 0.7 is plotted in Fig. 5. It
agrees well with other numerical simulations in literature [16, 19].
Example 4.6 (2-D linear diffusion equation)
Ut − (Uxx + Uyy ) = 0,
(x, y) ∈ (0, 2π) × (0, 2π),
(4.11)
with = 0.1 and initial condition as U (x, y, 0) = sin(x + y). We obtain optimal (k + 1)th
order of accuracy with P k polynomial approximations, see Table 6. Notice the optimal convergence is obtained with zero penalty coefficient β0 = β0u − β0v = 0 in the numerical flux,
which is quite an improvement compared to the Oden-Baumann scheme [5]. Comparing to
the NIPG method [22], similar optimal convergence is recovered for even order polynomial
approximations with positive penalty coefficient β0 = β0u − β0v > 0.
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Fig. 4 Nonlinear porous medium equation with m = 6 in (4.4) and initial profile (4.8). Two box collision
with different heights. Piecewise quadratic approximations with mesh N = 160
Fig. 5 Strongly degenerate convection diffusion equation (4.9). P 2 polynomial approximations. Left: initial
profile. Right: t = 0.7 with symbol circle for N = 100 and solid line for N = 600
Example 4.7 (2-D nonlinear porous medium equation)
Ut = U 2 xx + U 2 yy ,
(x, y) ∈ [−10, 10] × [−10, 10],
(4.12)
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681
with zero boundary condition and the following initial condition,
⎧
⎨ 1, (x − 2)2 + (y + 2)2 < 6,
U (x, 0) = 1, (x + 2)2 + (y − 2)2 < 6,
⎩
0, otherwise.
(4.13)
We compute the nonsymmetric DG solution with piecewise linear polynomial approximations and plot the results in Fig. 6 for t = 0, 0.1, 0.5, 2.0. Due to the nonlinearity of the given
problem, we see that second order derivative jump terms are included in the numerical flux
even with piecewise linear polynomial approximations. We observe the initial discontinuous
disks quickly smooth out, merge into each other with finite speed, and even out with time
evolution. The results agree well with those in literature [19].
Table 6 2-D linear diffusion equation (4.11). L2 and L∞ errors and orders at t = 0.5. P k polynomial
approximations with k = 2, 3, 4
k
k=2
k=3
k=4
β0
0
0
0
β1
1/12
1/12
1/40
N = 10 × 10
N = 20 × 20
N = 40 × 40
N = 80 × 80
Error
Error
Order
Error
Order
Error
Order
L2
2.90E-03
3.63E-04
3.0
4.35E-05
3.1
5.37E-06
3.0
L∞
2.26E-02
2.33E-03
3.3
2.92E-04
3.0
3.65E-05
3.0
L2
3.67E-04
2.13E-05
4.1
1.28E-06
4.0
7.98E-08
4.0
L∞
1.90E-03
1.15E-04
4.0
7.31E-06
4.0
4.58E-07
4.0
L2
1.46E-05
3.92E-07
5.2
1.16E-08
5.1
3.59E-10
5.0
L∞
1.78E-04
5.67E-06
5.0
1.78E-07
5.0
5.58E-09
5.0
Fig. 6 2-D porous medium equation (4.12). Mesh N × N = 100 × 100 at time t = 0, 0.1, 0.5 and 2.0
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Fig. 7 2-D degenerate parabolic
problem (4.14). Mesh
N × N = 200 × 200 at T = 0.5
Example 4.8 (2-D strongly degenerate parabolic problem)
Ut + U 2 x + U 2 y = ν(U )Ux x + ν(U )Uy y , (x, y) ∈ [−1.5, 1.5] × [−1.5, 1.5],
(4.14)
with zero boundary condition and initial condition,
⎧
(x + 0.5)2 + (y + 0.5)2 < 0.16,
⎨ 1,
U (x, y, 0) = −1, (x − 0.5)2 + (y − 0.5)2 < 0.16,
⎩
0,
otherwise.
= 0.1 and ν(U ) is as defined in Example 4.5. We compute the nonsymmetric DG solution
with piecewise linear polynomial approximations and plot the results in Fig. 7 for T = 0.5.
The results agree well with those in literature [16] and [19].
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