ENERGY STABILITY OF THE MUSCL SCHEME Qaisar Abbas and Jan Nordstr¨om

ENERGY STABILITY OF THE MUSCL SCHEME Qaisar Abbas and Jan Nordstr¨om
Seventh South African Conference on Computational and Applied Mechanics
SACAM10
Pretoria, 10−13 January 2010
c
⃝SACAM
ENERGY STABILITY OF THE MUSCL SCHEME
Qaisar Abbas∗,1 , Edwin van der Weide† and Jan Nordström∗,‡,2
∗
Department of Information Technology, Scientific Computing
Uppsala University, SE-751 05 Uppsala, Sweden,
1
[email protected]
†
Faculty of Engineering Technology
University of Twente, 7500 AE Enschede, The Netherlands,
[email protected]
‡
School of Mechanical, Industrial and Aeronautical Engineering
University of the Witvatersrand, PO WITS 2050, Johannesburg, South Africa, and
Department of Aeronautics and Systems Integration
FOI, The Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden,
2
[email protected]
Keywords: MUSCL Scheme, Energy Estimates and Numerical Stability, Summation-by-parts
Form, Artificial Dissipation
Abstract
We analyze the energy stability of the standard MUSCL scheme. The analysis is possible by
reformulating the MUSCL scheme in the framework of summation-by-parts (SBP) operators
including an artificial dissipation. The effect of different slope limiters is studied. It is found
that all the slope limiters do not lead to the correct sign of the entries in the dissipation matrix.
The implication of that is discussed for both linear and nonlinear scalar problems.
1
Introduction
For problems involving shocks which arise in computational fluid mechanics and related areas,
the MUSCL scheme [1] is a very effective approach to resolve discontinuities. This scheme
ensures the monotonicity of the solution for the whole computational time and it is arguably
computationally less expensive compared to relevant counterparts like the WENO schemes [2]
for approximately the same accuracy.
In this paper, we have reformulated the MUSCL scheme in summation-by-parts (SBP) form
including an artificial dissipation operator. Related work can be found in [3, 4], where the
WENO scheme has been formulated in a similar way. The SBP operators are well-established
theoretically [9] and their usefulness is proven for practical applications, see [10, 11, 12].
In this work we will investigate the MUSCL scheme to see if the scheme is energy stable, i.e.
stable in the L2 -norm, see [3, 4]. We consider both scalar linear and nonlinear hyperbolic problems in one dimension. Our analysis is based on theoretical as well as numerical observations.
2
The MUSCL Scheme in SBP Form
Consider the unsteady one-dimensional conservation law
0 ≤ x ≤ 1,
ut + f (u)x = 0,
t ≥ 0.
(1)
Define a uniform grid xj = j∆x, j = 0, . . . , N , with ∆x = 1/N . On the grid, define a flux
F (U ), where U = [U0 (t), U1 (t) . . . , UN (t)]T is the discrete approximation of the solution u
in Eq. (1). The second order upwind discretization of Eq. (1) using the MUSCL approach [1]
results in
)
1 (
Ut + RESi = 0, RESi =
F 1 − Fi− 1 .
(2)
2
∆x i+ 2
In Eq. (2), Fi+ 1 is the flux function at the interface i + 12 . More details on the computation of
2
numerical flux function can be found in [5].
Similarly a second order discretization of the flux function in Eq. (1), obeying the SBP property
[9] and with the introduction of artificial dissipation on SBP form [6] leads to
Ut + D2 F = −P −1 D̃1T BM D̃1 U,
(3)
where D2 is the central finite difference operator on SBP form given by
D2 = P −1 Q,

Q + QT = B,

−2 2
 −1 0
1
1 

.. .. ..
D2 =

.
.
.
2∆x 

−1 0 1
−2 2



,


P −1
P = PT,

2

1
1 

..
=

.
∆x 

1
(4)




,


2
and B = diag(−1, 0, . . . , 0, 1). The term −P −1 D̃1T BM D̃1 U in Eq. (3) is an artificial dissipation operator. It will be shown below that the matrix BM can be constructed such that Eq. (3)
corresponds to the standard second order MUSCL formulation [1] which means that the fore 1 is a two point difference operator and
mulations given by Eq. (2) and Eq. (3) are equivalent. D
the matrix BM is a diagonal matrix, see Eq. (5).




b0
0
−1 1




b1
−1 1







.
.
.
e1 = 
.
.
.
(5)
,
B
=
D
.



.
.
.
M







bN −1
−1 1 
0
0
−1 1
2.1
Explicit Form of BM
At an interior point i, we have
}
{
e 1 U = − 1 (bi−1 ∆Ui−1 − bi ∆Ui ),
e 1T BM D
−P −1 D
∆x
i
(6)
where ∆Ui = Ui+1 − Ui . In combination with the central discretization of the convective term,
this leads to the following formulation of the residual for an internal node
∆xRESi =
1
(Fi+1 − Fi−1 ) + bi−1 ∆Ui−1 − bi ∆Ui .
2
(7)
For the boundary nodes x0 and xN , the residuals are
∆xRES0 = ∆F0 − Pe0−1 b0 ∆U0 ,
∆xRESN = ∆FN −1 + PeN−1 bN −1 ∆UN −1 ,
(8)
where Pe0−1 = PeN−1 = 2. Comparing Eqs. (2) and (7), it is clear that both schemes are identical
if
1
bi ∆Ui = (Fi+1 + Fi ) − Fi+ 1 .
(9)
2
2
It can be shown that the bi in Eq. (9) becomes
{
)
}
(
1 ϕi ψi+1
ψi+1
ϕi
bi =
−
+ ARi+ 1 ,i+1
− ALi+ 1 ,i
,
(10)
Ai+ 12 1 −
2
2
2
2
2
2
2
where ϕi and ψi+1 are the slope limiters involved in the fluxes. They are related in the following
way.
(
)
( )
∆Ui−1
1
∆Ui−1
∆Ui−1
ϕi = ϕ (ri ) = ϕ
=
ϕ
=
ψi .
(11)
∆Ui
∆Ui
ri
∆Ui
∂F
Also A = ∂U
is a Jacobian matrix evaluated at the Roe average states. The property of a Roe
average state is that f2 − f1 = ARoe (u2 − u1 ).
2.2 Energy Stability
In this section we define the two versions of the energy stability, that we will work with in the
analysis below.
Definition. Consider Eq. (3) and Eq. (12). The scheme defined by Eq. (3) is pointwise energy
stable if bi ≥ 0 for all i = 0, 1 . . . , N . The scheme defined by Eq. (3) is energy stable in the
mean if (DU )T BM (DU ) ≥ 0, where DU = [(DU )0 , (DU )1 , . . . , (DU )N ]T .
Remark. Pointwise energy stable schemes lead to energy stable schemes in the mean. The
reverse is not true.
2.3 Energy Estimates
To investigate whether the scheme defined in Eq. (3) is energy stable or not, we start by considering the linear constant coefficient case with F = aU and use the energy method. Multiplying
Eq. (3) with U T P , adding its transpose and using Eq. (4) leads to
d
e 1 U )T BM (D
e 1 U ).
||U ||2P + aU T BU = −2(D
dt
(12)
where ||U ||2P = U T P U . For a bounded solution and energy stability we must have dtd ||U ||2P ≤
0. The boundary terms U T BU = U02 − UN2 can be bounded using the SAT boundary treatment
[8] and are ignored from now on. The right-hand side of Eq. (12) is negative if the matrix BM
is positive semi-definite. The matrix BM for a linear problem becomes
bi =
}
1{
|A| − ϕi A+ + ψi+1 A− ,
2
(13)
where A+ contains the positive eigenvalues of A and A− the negative ones,
A+ =
1
(A + |A|) ,
2
1
(A − |A|) .
2
A− =
(14)
For a scalar problem with F = aU , Eq. (13) reduces to
1
bi = a {1 − ϕi } ,
2
a > 0,
and
bi =
1
|a| {1 − ψi+1 } ,
2
a < 0.
(15)
From the theory of the slope limiters [7] we have that 0 ≤ ϕi , ψi+1 ≤ 2. It is obvious that
any limiter which takes values greater than 1, will lead to bi ≤ 0 in the computational domain
and hence no pointwise energy stability. In [3, 4], the authors modified the WENO scheme by
correcting this anomaly of the scheme. We will discuss below whether that is necessary and
meaningful.
3
Numerical Results
Consider Eqs. (10), (11) and (12). It is obvious that the sign of bi depends on the slope limiters
involved in the MUSCL scheme. If the solution is smooth, we have ϕi = ψi+1 = 1, and for all
A, bi will be zero. For problems with discontinuities , we could have 0 ≤ ϕi , ψi+1 ≤ 2, which
decides the sign of bi in non-smooth regions.
We consider a linear problem (f = u) first with a step discontinuity as initial data and analyze
four different limiters. All the results are shown for N = 80 and t = 0.3. In Figures 1–4
we have shown the minimum of bi and −(D1 U )T BM (D1 U ) at each time step for minmod,
VanLeer, superbee and MC limiters. The minmod limiter have bi ≥ 0 for all time and hence is
pointwise stable. All other limiters lead to bi < 0 at few points near the discontinuity. It means
that these limiters do not lead to pointwise stability. It is also found that −(D1 U )T BM (D1 U ) ≤
0 for all limiters for the whole computational time which gives energy stability in the mean.
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
(a) min(bi ) per time step
200
2
Initial soluton
Numerical solution
Exact solution
−0.1
solution vector
dissipation / time step
min(bi) / time step
1
−0.2
−0.3
−0.4
1.5
1
−0.5
−0.6
0
50
100
150
Number of time steps
(b) −(D1 U )T BM (D1 U )
200
0.5
0
0.2
0.4
x
0.6
0.8
1
(c) solution, N = 80, t = 0.3
Figure 1: Results from the MUSCL in SBP form using the minmod limiter, f = u.
2
Next we consider the Burger equation with f = u2 in Eq. (1) and repeat the same analysis
with the minmod, VanLeer and MC limiters. It is found that all the tested limiters have some
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
−0.2
−0.3
−0.4
50
100
150
Number of time steps
1.5
1
0.5
0
200
(b) −(D1 U )T BM (D1 U )
(a) min(bi ) per time step
Initial soluton
Numerical solution
Exact solution
−0.5
−0.6
0
200
2
−0.1
solution vector
dissipation / time step
min(bi) / time step
1
0.2
0.4
x
0.6
0.8
1
(c) solution, N = 80, t = 0.3
Figure 2: Results from the MUSCL in SBP form using the Van Leer limiter, f = u.
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
−0.2
−0.3
−0.4
1.5
1
−0.5
50
100
150
Number of time steps
0.5
0
200
(b) −(D1 U )T BM (D1 U )
(a) min(bi ) per time step
Initial soluton
Numerical solution
Exact solution
−0.1
−0.6
0
200
2
solution vector
dissipation / time step
min(bi) / time step
1
0.2
0.4
x
0.6
0.8
1
(c) solution, N = 80, t = 0.3
Figure 3: Results from the MUSCL in SBP form using the Superbee limiter, f = u.
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
−0.2
−0.3
−0.4
50
100
150
Number of time steps
1.5
1
0.5
0
200
(b) −(D1 U )T BM (D1 U )
(a) min(bi ) per time step
Initial soluton
Numerical solution
Exact solution
−0.5
−0.6
0
200
2
−0.1
solution vector
dissipation / time step
min(bi) / time step
1
0.2
0.4
x
0.6
0.8
1
(c) solution, N = 80, t = 0.3
Figure 4: Results from the MUSCL in SBP form using the MC limiter, f = u.
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
(a) min(bi ) per time step
200
2
Initial soluton
Numerical solution
Exact solution
−0.1
solution vector
dissipation / time step
min(bi) / time step
1
−0.2
−0.3
−0.4
1.5
1
−0.5
−0.6
0
50
100
150
Number of time steps
(b) −(D1 U )T BM (D1 U )
200
0.5
0
0.2
0.4
x
0.6
0.8
(c) solution, N = 80, t = 0.3
Figure 5: Results from the MUSCL in SBP form using the minmod limiter, f =
u2
2 .
1
0
0.5
0
−0.5
−1
0
50
100
150
Number of time steps
−0.2
−0.3
−0.4
50
100
150
Number of time steps
1.5
1
0.5
0
200
(b) −(D1 U )T BM (D1 U )
(a) min(bi ) per time step
Initial soluton
Numerical solution
Exact solution
−0.5
−0.6
0
200
2
−0.1
solution vector
dissipation / time step
min(bi) / time step
1
0.2
0.4
0
−0.5
−1
0
50
100
150
Number of time steps
2
−0.2
−0.3
−0.4
50
100
150
Number of time steps
Initial soluton
Numerical solution
Exact solution
1.5
1
0.5
0
200
(b) −(D1 U )T BM (D1 U )
(a) min(bi ) per time step
0.2
0.4
0
−0.5
−1
0
50
100
150
Number of time steps
(a) min(bi ) per time step
200
0.6
0.8
1
u2
2 .
2
Initial soluton
Numerical solution
Exact solution
−0.1
solution vector
dissipation / time step
min(bi) / time step
0
0.5
x
(c) solution, N = 80, t = 0.3
Figure 7: Results from the MUSCL in SBP form using the Superbee limiter, f =
1
1
−0.5
−0.6
0
200
0.8
u2
2 .
−0.1
solution vector
dissipation / time step
min(bi) / time step
0
0.5
0.6
(c) solution, N = 80, t = 0.3
Figure 6: Results from the MUSCL in SBP form using the Van Leer limiter, f =
1
x
−0.2
−0.3
−0.4
1.5
1
−0.5
−0.6
0
50
100
150
Number of time steps
(b) −(D1 U )T BM (D1 U )
200
0.5
0
0.2
0.4
x
0.6
0.8
1
(c) solution, N = 80, t = 0.3
Figure 8: Results from the MUSCL in SBP form using the MC limiter, f =
u2
2 .
bi < 0 but the minmod limiter is almost zero for all time leading to pointwise stability, see
Figures 5–8. It can also be seen that all schemes are stable in the mean.
It is not clear whether pointwise stability is necessary or if stability in the mean is enough. If
we replace bi < 0 in the matrix BM with bi = 0 at each time step, we find that it leads to an
additional and excessive amount of dissipation in the discontinuity/shock region, see Table 1
for l2 -error of solutions. By demanding the pointwise stability, clearly the sharpness of the
shock decreases.
Table 1: Analysis of different limiters for the linear problem (a = 1, t = 0.3)
4
Limiters
l2 -error, min(bi ≤ 0)
l2 -error, min(bi = 0)
minmod
Van Leer
Superbee
MC
0.0711
0.0578
0.0400
0.0504
0.0711
0.0642
0.0634
0.0639
Conclusion
We have expressed the MUSCL scheme as a combination of an SBP operator and an artificial
dissipation operator. This form allows us to use the energy method for analyzing stability. Our
main interest was to look at the behavior of dissipation matrix BM in Eq. (5), which is crucial
for the stability of the scheme and also influence the sharpness of the shock.
As the matrix depends on the slope limiters of the MUSCL scheme, it was found most of the
tested limiters except minmod limiter do not lead to pointwise stability while all limiters are
stable in the mean.
By making the schemes pointwise stable by replacing bi < 0 in the matrix BM with bi = 0
resulted in an additional and excessive dissipation for all the limiters. It was shown that the
error in the calculations increased and the sharpness of the shock decreased. This procedure
was used in [3, 4] but seems questionable.
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