-
High-order Runge-Kutta discontinuous Galerkin
methods with multi-resolution WENO limiters for
solving steady-state problems
Jun Zhu1, Chi-Wang Shu2 and Jianxian Qiu3
Abstract
In this paper, we design a new troubled cell indicator, applying
high-order finite vol-
ume multi-resolution weighted essentially non-oscillatory (WENO)
techniques to serve as
limiters for high-order Runge-Kutta discontinuous Galerkin
(RKDG) methods in simulat-
ing steady-state problems and pushing the residue to settle down
close to machine zero on
structured meshes. Firstly, a new troubled cell indicator is
designed to precisely detect the
cells which would need further limiting procedures. Then the
arbitrary high-order multi-
resolution WENO limiting procedures are adopted by using
information of the DG solution
essentially only within the troubled cell itself, to build a
sequence of hierarchical L2 projec-
tion polynomials from zeroth degree to the highest degree of the
RKDG methods. These
RKDG methods with multi-resolution WENO limiters could use the
same compact spatial
stencil as that of the original RKDG methods, could maintain the
originally designed high
order accuracy in smooth regions (verified from second-order to
fifth-order as examples),
and could gradually degrade to first-order so as to suppress
slight post-shock oscillations
near strong discontinuities when computing steady-state
problems. The linear weights in
the multi-resolution WENO limiting procedures can be any
positive numbers on the condi-
tion that their sum equals one. These new multi-resolution WENO
limiters are very simple
to construct, and can be easily implemented to arbitrary
high-order accuracy for solving
steady-state problems in multi-dimensions.
Key Words: multi-resolution WENO limiter, RKDG method, slight
post-shock oscilla-
tion, machine zero, steady-state problem.
AMS (MOS) subject classification: 65M60, 35L65
1College of Science, Nanjing University of Aeronautics and
Astronautics, Nanjing, Jiangsu 210016, P.R.
China. E-mail: [email protected]. Research was supported by
NSFC grant 11872210 and Science Chal-
lenge Project, No. TZ2016002. The author was also partly
supported by NSFC grant 11926103 when he
visited Tianyuan Mathematical Center in Southeast China, Xiamen,
Fujian 361005, P.R. China.2Division of Applied Mathematics, Brown
University, Providence, RI 02912, USA. E-mail: chi-
wang [email protected]. Research was supported by AFOSR grant
FA9550-20-1-0055 and NSF grant DMS-
2010107.3School of Mathematical Sciences and Fujian Provincial
Key Laboratory of Mathematical Modeling and
High-Performance Scientific Computing, Xiamen University,
Xiamen, Fujian 361005, P.R. China. E-mail:
[email protected]. Research was supported by NSAF grant U1630247
and Science Challenge Project, No.
TZ2016002.
1
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1 Introduction
In this paper, high-order Runge-Kutta discontinuous Galerkin
(RKDG) methods [7, 8, 9,
11] with new multi-resolution WENO limiters [48] are applied to
solve steady Euler equations
{
f(u)x + g(u)y = 0,u(x, y) = u0(x, y),
(1.1)
on structured meshes. One way to get a numerical solution of
(1.1) is to solve the associated
unsteady Euler equations{
ut + f(u)x + g(u)y = 0,u(x, y, 0) = u0(x, y),
(1.2)
and then drive the residue to zero. High-order DG methods are
applied to discretize the
spatial variables and explicit, nonlinearly stable high-order
Runge-Kutta methods [40, 12]
are adopted to discretize the temporal variable. Our main
objective of this paper is to
design a new troubled cell indicator to precisely detect the
cells that need further limiting
procedures and then adopt the arbitrary high-order spatial
limiting procedures [48] (the
second-order, third-order, fourth-order, and fifth-order
versions are taken as examples) for
the RKDG methods to solve two-dimensional steady-state
problems.
If one confirms that the residue of the unsteady Euler equations
(1.2) is small enough,
ideally at or close to the level of machine zero, the numerical
solution of the steady Euler
equations (1.1) is acceptable. The appearance of strong
discontinuities in the simulation
of (1.1) and (1.2) is the main difficulty. If the numerical
solution has strong shocks or
contact discontinuities, its physical variables change abruptly.
Some high-order schemes
cannot suppress oscillations near strong discontinuities. Many
high-resolution or high-order
numerical schemes have been designed with the aim of controlling
the oscillations by the
use of artificial viscosities [24, 25] or limiters [18, 24, 41],
respectively. The application of
artificial viscosity results in a method to be easily
implemented, and its residue can often
converge close to machine zero. Jameson et al. [23, 26] proposed
a third-order finite volume
discretization method with dissipative terms and applied a
Runge-Kutta time discretization
method for solving the steady Euler equations. However, the main
drawback of such schemes
2
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is that one often needs to adjust certain parameters in the
artificial viscosity to maintain
sharp shock transitions and to suppress oscillations near strong
shocks. If limiters are used
in designing numerical schemes, such numerical schemes could be
very efficient in computing
supersonic flows including strong shocks and contact
discontinuities [18]. Yet the application
of total variation diminishing (TVD) type limiters will degrade
the accuracy of the numerical
scheme to first-order near local smooth extrema [35], and the
lack of sufficient smoothness
of the numerical fluxes with the application of such limiters
often results in the residue
not converging close to machine zero. Yee et al. [44] designed
an implicit stable high-
resolution TVD scheme and applied it to compute steady-state
problems. Yee and Harten
[43] designed TVD schemes to solve multi-dimensional hyperbolic
conservation laws and
steady-state problems in curvilinear coordinates.
Many high-resolution or high-order schemes have been designed to
improve the first-
order methods [17] to arbitrary high-order accuracy for solving
unsteady problems. Harten
et al. introduced essentially non-oscillatory (ENO) schemes to
obtain uniform high-order
accuracy, and applied finite volume ENO schemes to compute
unsteady problems [20]. Such
finite volume ENO schemes apply the locally smoothest spatial
stencil and abandon all the
others when approximating the variables at cell boundaries,
resulting in high-order accuracy
in smooth regions and suppressing oscillations in nonsmooth
regions. Later, Shu et al. de-
signed finite difference ENO schemes with a TVD Runge-Kutta time
discretization [40] for
multi-dimensional computation. In 1994, Liu et al. [32] designed
a high-order (third-order
as an example) finite volume weighted ENO (WENO) scheme using a
convex combination
of the same candidate spatial stencils of an r-th order ENO
scheme, to obtain an (r+1)-th
order accuracy in smooth regions. In 1996, Jiang and Shu [27]
first designed a finite dif-
ference WENO scheme from the the same candidate stencils of an
r-th order ENO scheme
to obtain a (2r-1)-th order scheme in smooth regions which can
suppress oscillations in
nonsmooth regions. Hereafter, two-dimensional finite volume WENO
schemes [15, 22] and
three-dimensional finite volume WENO schemes [47] were designed
on unstructured meshes.
3
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It has been observed that a high-order WENO-type spatial
reconstruction procedure with
a high-order TVD Runge-Kutta time discretization method [40]
could obtain good numer-
ical results for solving unsteady problems containing all kinds
of smooth structures, strong
shocks, and contact discontinuities. When the classical
high-order WENO schemes [27] are
used to solve for the steady-state problems, their residue often
hangs at a truncation error
level without settling down close to machine zero even after a
long time iteration. Serna et al.
[39] proposed a new limiter to reconstruct the numerical flux
and improve the convergence of
the numerical solution to steady states. Zhang et al. [46] found
that slight post-shock oscil-
lations would propagate from the region near the shocks
downstream to the smooth regions
and result in the residue hanging at a high truncation error
level rather than converging
to machine zero. Zhang et al. [45] designed an upwind-biased
interpolation technique to
improve the convergence of high-order WENO scheme for
steady-state problems. But the
residue computed by such new schemes still could not converge
close to machine zero for
some two-dimensional steady-state problems [45]. In 2016, a
novel high-order fixed-point
sweeping WENO method [42] was proposed to simulate steady-state
problems and could
obtain better convergence property. However, the residue could
not settle down close to
machine zero for some benchmark steady-state tests as
before.
Now let us first review the history of the development of
discontinuous Galerkin (DG)
methods for solving unsteady problems. In 1973, Reed and Hill
[38] designed the first DG
method in the framework of neutron transport. Due to its
desirable properties, DG methods
were also used extensively in atmospheric science [34]. The
reconstruction operator [13] was
applied at the beginning of each time step in the computation to
increase the formal order
of accuracy of high-order DG methods. A novel weighted RKDG
method [21] was designed
for three-dimensional acoustic and elastic wave, and
reconstructed DG (rDG) methods [33]
were proposed for diffusion equations. Other high-order DG
methods can be found in [29]. If
unsteady or steady-state problems are not smooth enough, their
numerical solutions might
contain oscillations near strong discontinuities and result in
nonlinear instability in non-
4
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smooth regions. One possible methodology to suppress
oscillations is to apply nonlinear
limiters to the high-order RKDG methods. A major development of
the DG method with a
classical minmod type total variation bounded (TVB) limiter was
carried out by Cockburn et
al. in a series of papers [7, 8, 9, 10, 11] to solve nonlinear
time dependent hyperbolic conser-
vation laws with an explicit, nonlinearly stable high-order
Runge-Kutta time discretization
method [40]. From then on, such methods are termed as RKDG
methods. One type of
limiters is based on slope modification, such as classical
minmod type limiters [7, 8, 9, 11],
the moment based limiter [1], and an improved moment limiter
[4]. Such limiters belong to
the slope type limiters and they could suppress oscillations at
the price of possibly degrading
numerical accuracy at smooth extrema. Another type of limiters
is based on the essentially
non-oscillatory (ENO) and weighted ENO (WENO) methodologies [15,
22, 27, 32], which
can achieve high-order accuracy in smooth regions and keep
essentially non-oscillatory prop-
erty near strong discontinuities. These WENO limiters are
basically designed in a finite
volume WENO fashion, but they need a wider spatial stencil for
high-order schemes. The
WENO limiters [37], central WENO (CWENO) limiters [3], and
Hermite WENO limiters
[36] belong to the second type of limiters. Since CWENO schemes
[30] are computationally
less expensive than the classical WENO reconstruction algorithms
[14], they can serve as a
posteriori subcell limiters for DG schemes [34]. However, it is
very difficult to implement
RKDG methods with the applications of WENO limiters, CWENO
limiters, or Hermite
WENO limiters for solving steady-state problems on structured or
unstructured meshes.
When such high-order RKDG methods are applied to compute steady
Euler equations, the
residual could not converge close to machine zero and would hang
at a higher truncation
error level.
More recently, a new type of high-order multi-resolution WENO
schemes has been de-
signed to solve time dependent hyperbolic conservation laws on
structured meshes [49]. We
design this new type of multi-resolution WENO schemes borrowing
the idea of the multi-
resolution methods [19]. For the purpose of designing finite
difference or finite volume multi-
5
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resolution WENO schemes, we only use the point values or cell
averages of the numerical
solution on a hierarchy of nested central spatial stencils, and
do not introduce any equiva-
lent multi-resolution representations. These new
multi-resolution WENO schemes adopt the
same largest stencil and apply a smaller number of stencils in
designing high-order spatial
approximation procedures than that of the classical WENO schemes
in [22, 47] on triangular
meshes or tetrahedral meshes, could obtain the optimal order of
accuracy in smooth regions,
and could gradually degrade from the optimal order to
first-order accuracy near strong dis-
continuities. In this paper, we extend high-order RKDG methods
with arbitrary high-order
multi-resolution WENO limiters [48] from solving unsteady Euler
equations to steady Euler
equations with the application of a new troubled cell indicator
on structured meshes. This
new troubled cell indicator is very simple and works well for
precisely detecting the cells
that need further limiting procedure. To the best of our
knowledge, it is the first type of
high-order RKDG methods with WENO limiters that could confirm
the residue to converge
close to machine zero for two-dimensional steady-state problems
with the application of a
classical third-order Runge-Kutta time discretization method
[40]. Of course, other time
marching methods as well as special tools such as
preconditioning to speed up steady-state
convergence could make the steady-state convergence more
efficient, however this is not the
focus of the current paper and hence will not be further
explored.
This paper is organized as follows. In Section 2, we give a
brief review of the RKDG
methods, propose a new troubled cell indicator to detect the
cells needing further limiting
procedures, and then design arbitrary high-order limiting
procedures using second-order,
third-order, fourth-order, and fifth-order multi-resolution WENO
limiters for steady-state
computations as examples. In Section 3, several standard
steady-state problems including
sophisticated wave structures, both inside the computational
fields and passing through the
boundaries of the computational domain, are presented to
demonstrate the good performance
of residue convergence close to machine zero. Concluding remarks
are given in Section 4.
6
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2 RKDG methods with multi-resolution WENO lim-
iters for steady-state computation
In this section, we first give a brief review of the RKDG
methods for solving (1.2).
The two-dimensional computational domain is divided by
rectangular cells Ii,j = Ii × Jj =
[xi− 12
, xi+ 12
]× [yj− 12
, yj+ 12
], i = 1, · · · , Nx and j = 1, · · · , Ny with the cell sizes
xi+ 12
− xi− 12
=
∆xi, yj+ 12
− yj− 12
= ∆yj, and cell centers (xi, yj) = (12(xi+ 1
2
+ xi− 12
), 12(yj+ 1
2
+ yj− 12
)). The
function space is defined by W kh = {v(x, y) : v(x, y)|Ii,j ∈
Pk(Ii,j)} as the piecewise poly-
nomial space of degree at most k defined on Ii,j. We also adopt
similar local orthonormal
basis over Ii,j, {v(i,j)l (x, y), l = 0, 1, ..., K; K =
(k+1)(k+2)2
− 1} as specified in [48]. The
two-dimensional solution uh(x, y, t) ∈ Wkh can be written
as:
uh(x, y, t) =
K∑
l=0
u(l)i,j(t)v
(i,j)l (x, y), x ∈ Ii,j, (2.1)
and the degrees of freedom u(l)i,j(t) are the moments defined
by
u(l)i,j(t) =
1
∆xi∆yj
∫
Ii,j
uh(x, y, t)v(i,j)l (x, y)dxdy, l = 0, ..., K. (2.2)
In order to determine the approximation solution, we evolve the
degrees of freedom u(l)i,j(t):
ddtu(l)i,j(t) =
1∆xi∆yj
(
∫
Ii,j
(
f(uh(x, y, t))∂∂xv(i,j)l (x, y) + g(uh(x, y, t))
∂∂yv(i,j)l (x, y)
)
dxdy
−∫
Ij
(
f̂(uh(xi+ 12
, y, t))v(i,j)l (xi+ 1
2
, y)− f̂(uh(xi− 12
, y, t))v(i,j)l (xi− 1
2
, y))
dy
−∫
Ii
(
ĝ(uh(x, yj+ 12
, t))v(i,j)l (x, yj+ 1
2
)− ĝ(uh(x, yj− 12
, t))v(i,j)l (x, yj− 1
2
))
dx)
,
l = 0, ..., K,(2.3)
where the ”hat” terms are the numerical fluxes (f̂ and ĝ are
monotone fluxes for the scalar
case and exact or approximate Riemann solvers for the system
case). The integrals in (2.3)
are computed by applying suitable numerical quadratures. The
semi-discrete scheme (2.3)
can be discretized in time by a third-order TVD Runge-Kutta time
discretization method
[40]:
u(1) = un +∆tL(un),u(2) = 3
4un + 1
4u(1) + 1
4∆tL(u(1)),
un+1 = 13un + 2
3u(2) + 2
3∆tL(u(2)).
(2.4)
7
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In order to explain how to apply a nonlinear limiter for the
RKDG methods, we adopt a
forward Euler time discretization of (2.3) as an example.
Starting from a solution unh ∈ Wkh
at time level n, we limit it to obtain a new function un,new
before advancing it to the next
time level. We need to find un+1h ∈ Wkh which satisfies
∫
Ii,j
un+1h
−un,newh
∆tv dxdy −
∫
Ii,j(f(un,newh )vx + g(u
n,newh )vy) dxdy
+∫
Ij
(
f̂(un,newh |x=xi+12
)v(x−i+ 1
2
, y)− f̂(un,newh |x=xi−12
)v(x+i− 1
2
, y))
dy
+∫
Ii
(
ĝ(un,newh |y=yj+12
)v(x, y−j+ 1
2
)− ĝ(un,newh |y=yj− 12
)v(x, y+j− 1
2
))
dx = 0,
(2.5)
for all test functions v(x, y) ∈ W kh . We will narrate how to
obtain the two-dimensional
un,newh |Ii,j in details. For simplicity, we omit the
sup-indices in un,newh |Ii,j , if it does not cause
confusion in the following.
First of all, we design a new troubled cell indicator to detect
the cells that may contain
strong discontinuities and in which the multi-resolution WENO
limiter is applied. Other
trouble cell detectors can of course also be used for solving
unsteady problems, but many of
them do not work well in solving steady-state problems,
according to our experiments. In
two-dimensional steady-state cases, we define the cell Ii,j to
be a troubled cell when
maxIℓ∈{Ii±1,j ,Ii,j±1}
(∣
∣
∣
∫
Iℓuh|Iℓdxdy −
∫
Ii,juh|Ii,jdxdy
∣
∣
∣
)
hi,j ·minIℓ∈{Ii±1,j ,Ii,j±1,Ii,j}
(∣
∣
∣
∫
Iℓuh|Iℓdxdy
∣
∣
∣
) ≥ Ck, (2.6)
where hi,j is the radius of the circumscribed circle in cell
Ii,j and Ck is a constant, usually,
we take Ck = 1 as specified in [28]. By using (2.6), we do not
need to adopt different
values of Ck to compute multi-dimensional problems as in [16]
and can simply set Ck = 1
for computing two-dimensional steady-state problems. This new
troubled cell indicator is
simple and robust enough in simulating steady-state problems
without identifying excessive
troubled cells inside the computational field. We emphasize our
observation that many other
troubled cell indicators [7, 8, 9, 10, 11, 28, 36, 37, 48] are
not good at precisely detecting
troubled cells which need further limiting procedures for
solving steady-state problems and
result in the residue to hang at a truncation error level
instead of converging close to machine
zero.
8
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Hereafter, we give details of the multi-resolution WENO limiter
for two-dimensional
scalar case. The crucial thought is to reconstruct a new
polynomial on the troubled cell
Ii,j which is a convex combination of polynomials of different
degrees: the DG solution
polynomial on this cell and a sequence of hierarchical
“modified” solution polynomials based
on the L2 projection methodology. For simplicity, we also
rewrite uh(x, y, t) to be uh(x, y) ∈
W kh in the following, if it does not cause confusion. A series
of unequal degree polynomials
qℓ(x, y), ℓ = 0, ..., k are constructed on the troubled cell
Ii,j:
∫
Ii,j
qℓ(x, y)v(i,j)l (x, y)dxdy =
∫
Ii,j
uh(x, y)v(i,j)l (x, y)dxdy, l = 0, ...,
(ℓ+ 1)(ℓ+ 2)
2− 1. (2.7)
Then we obtain equivalent expressions for these constructed
polynomials of different degrees.
To keep consistent notation, we will denote p0,1(x, y) = q0(x,
y). Following original ideas for
classical CWENO schemes [5, 30, 31], we obtain polynomials
pℓ,ℓ(x, y), ℓ = 1, ..., k through
pℓ,ℓ(x, y) =1
γℓ,ℓqℓ(x, y)−
γℓ−1,ℓγℓ,ℓ
pℓ−1,ℓ(x, y), ℓ = 1, ..., k, (2.8)
with γℓ−1,ℓ + γℓ,ℓ = 1 and γℓ,ℓ 6= 0, together with polynomials
pℓ,ℓ+1(x, y), ℓ = 1, ..., k − 1
through
pℓ,ℓ+1(x, y) = ωℓ,ℓpℓ,ℓ(x, y) + ωℓ−1,ℓpℓ−1,ℓ(x, y), ℓ = 1, ...,
k − 1, (2.9)
with ωℓ−1,ℓ+ωℓ,ℓ = 1. In (2.8), the γ’s are the linear weights
and we choose them as γℓ−1,ℓ =
0.01 and γℓ,ℓ = 0.99 for the numerical computations of all
steady-state problems. In (2.9),
the ω’s are the nonlinear weights which will be defined later.
The smoothness indicators βℓ,ℓ2
are computed by using the same recipe as in [27]:
βℓ,ℓ2 =
κ∑
|α|=1
∫
Ii,j
(∆xi∆yj)|α|−1
(
∂|α|
∂xα1∂yα2pℓ,ℓ2(x, y)
)2
dx dy, ℓ = ℓ2 − 1, ℓ2; ℓ2 = 1, ..., k,
(2.10)
where κ = ℓ, α = (α1, α2), and |α| = α1 + α2, respectively. The
only exception is β0,1 [48]:
we define qi,j−1(x, y) =∑2
l=0 u(l)i,j−1(t)v
(i,j−1)l (x, y), qi,j+1(x, y) =
∑2l=0 u
(l)i,j+1(t)v
(i,j+1)l (x, y),
qi−1,j(x, y) =∑2
l=0 u(l)i−1,j(t)v
(i−1,j)l (x, y), and qi,j+1(x, y) =
∑2l=0 u
(l)i,j+1(t)v
(i,j+1)l (x, y), respec-
9
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tively. Then the associated smoothness indicators are
ςi,j−1 =
∫
Ii,j
(
∂
∂xqi,j−1(x, y)
)2
+
(
∂
∂yqi,j−1(x, y)
)2
dxdy, (2.11)
ςi,j+1 =
∫
Ii,j
(
∂
∂xqi,j+1(x, y)
)2
+
(
∂
∂yqi,j+1(x, y)
)2
dxdy, (2.12)
ςi−1,j =
∫
Ii,j
(
∂
∂xqi−1,j(x, y)
)2
+
(
∂
∂yqi−1,j(x, y)
)2
dxdy, (2.13)
and
ςi+1,j =
∫
Ii,j
(
∂
∂xqi+1,j(x, y)
)2
+
(
∂
∂yqi+1,j(x, y)
)2
dxdy. (2.14)
After that, β0,1 is defined as
β0,1 = min(ςi,j−1, ςi,j+1, ςi−1,j , ςi+1,j). (2.15)
We adopt the WENO-Z recipe as shown in [2, 6] with
τℓ2 = (βℓ2,ℓ2 − βℓ2−1,ℓ2)2 , ℓ2 = 1, ..., k, (2.16)
to compute the nonlinear weights as
ωℓ1,ℓ2 =ω̄ℓ1,ℓ2
∑ℓ2ℓ=ℓ2−1
ω̄ℓ,ℓ2, ω̄ℓ1,ℓ2 = γℓ1,ℓ2
(
1 +τℓ2
ε+ βℓ1,ℓ2
)
, ℓ1 = ℓ2 − 1, ℓ2; ℓ2 = 1, ..., k. (2.17)
In this paper, ε is taken as 10−6 in all simulations of
steady-state problems. Finally, the new
reconstruction polynomial is defined as
unewh |Ii,j =
ℓ2∑
ℓ=ℓ2−1
ωℓ,ℓ2pℓ,ℓ2(x, y), ℓ2 = 1, ..., k, (2.18)
for obtaining (k+1)th-order spatial approximation. The scalar
multi-resolution WENO lim-
iting procedure can be easily extended to two-dimensional
systems as in [48], the details are
omitted here to save space.
10
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3 Numerical tests
In this section, we perform numerical experiments to test the
steady-state computation
performance of high-order RKDG methods with multi-resolution
WENO limiters described
in the previous section. The CFL number is 0.3 for the
second-order (P 1), 0.18 for the third-
order (P 2), 0.1 for the fourth-order (P 3), and 0.08 for the
fifth-order (P 4) RKDG methods,
respectively. For solving two-dimensional steady-state problems,
the time step is chosen
according to the CFL condition
∆tmax1≤i≤N
(
|µi|+ cihi
+|νi|+ ci
hi
)
≤ CFL,
in which µi is x-directional velocity, νi is y-directional
velocity, ci =√
γ piρi, hi is the diameter
of the inscribed circle of the target cell, and N is the total
number of the cells. The single
index i is used here to list all cells in the computational
field. Then the average residue is
defined as
ResA =N∑
i=1
|R1i|+ |R2i|+ |R3i|+ |R4i|
4×N, (3.1)
where R∗i are local residuals of different cell averages of the
conservative variables, that
is, R1i =∂ρ
∂t|i ≈
ρn+1i −ρni
∆t, R2i =
∂(ρµ)∂t
|i ≈(ρµ)n+1i −(ρµ)
ni
∆t, R3i =
∂(ρν)∂t
|i ≈(ρν)n+1i −(ρν)
ni
∆t, and
R4i =∂E∂t|i ≈
En+1i −Eni
∆t, respectively. All cells are set to be troubled cells in
Example 4.1,
so as to test numerical accuracy when the new type of
multi-resolution WENO limiting
procedure is enacted in the whole computational field. Then we
set the constant Ck in (2.6)
to be 1 in other steady-state problems.
Example 3.1. In this accuracy example, we study two-dimensional
Euler equations
∂
∂t
ρρµρνE
+∂
∂x
ρµρµ2 + pρµν
µ(E + p)
+∂
∂y
ρνρµν
ρν2 + pν(E + p)
= 0, (3.2)
with the exact steady-state solutions given by (1) ρ(x, y,∞) =
1+0.2 sin(x−y), µ(x, y,∞) =
1, ν(x, y,∞) = 1, and p(x, y,∞) = 1; (2) ρ(x, y,∞) = 1 + 0.2
sin(2(x − y)), µ(x, y,∞) =
11
-
Table 3.1: 2D Euler equations. Case (1). RKDG methods with
multi-resolution WENOlimiters. Steady state. L1 and L∞ errors.
Second-order method Third-order methodGrid cells L1 error order
L∞ error order L1 error order L∞ error order20×20 9.04E-5 9.42E-4
3.43E-6 2.29E-530×30 4.01E-5 2.00 4.22E-4 1.98 1.02E-6 2.98 7.00E-6
2.9240×40 2.25E-5 2.00 2.38E-4 1.99 4.33E-7 2.99 2.98E-6 2.9650×50
1.44E-5 2.00 1.52E-4 1.99 2.22E-7 2.99 1.53E-6 2.9860×60 1.00E-5
2.00 1.06E-4 1.99 1.28E-7 2.99 8.91E-7 2.99
Fourth-order method Fifth-order methodGrid cells L1 error order
L∞ error order L1 error order L∞ error order20×20 2.46E-8 2.59E-7
4.29E-10 3.08E-930×30 4.81E-9 4.03 5.12E-8 4.00 5.58E-11 5.03
4.10E-10 4.9740×40 1.51E-9 4.02 1.62E-8 4.00 1.31E-11 5.02 9.77E-11
4.9850×50 6.17E-10 4.02 6.64E-9 4.00 4.31E-12 5.01 3.21E-11
4.9860×60 2.96E-10 4.01 3.20E-9 4.00 1.74E-12 4.97 1.29E-11
4.97
1, ν(x, y,∞) = 1, and p(x, y,∞) = 1. We take the numerical
initial conditions as the
exact solution projected onto the grid, and then march to
numerical steady states. The
computational domain is (x, y) ∈ [0, 2] × [0, 2], and the exact
steady-state solutions are
applied as boundary conditions in both directions. The
convergence history of the residue
(3.1) as a function of time is shown in Figure 3.1 and Figure
3.2, in which we can see that
the residue settles down to tiny numbers close to machine zero.
The L1 and L∞ errors and
orders of accuracy at steady state are listed in Table 3.1 and
Table 3.2, from which we can
see that the designed second-order, third-order, fourth-order,
and fifth-order accuracies are
achieved for RKDG methods with multi-resolution WENO
limiters.
Example 3.2. Shock reflection problem. The computational domain
is a rectangle of length
4 and height 1. The boundary conditions are that of a reflection
condition along the bottom
boundary, supersonic outflow along the right boundary, and
Dirichlet conditions on the other
two sides:
(ρ, µ, ν, p)T ) =
{
(1.0, 2.9, 0, 1.0/1.4)T |(0,y,t)T ,(1.69997, 2.61934,−0.50632,
1.52819)T |(x,1,t)T .
12
-
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5-16
-14
-12
-10
-8
-6
-4
-2
1234
5
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5-16
-14
-12
-10
-8
-6
-4
-2
12345
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5
-12
-10
-8
1
2
3
4
5
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5
-12
-11
-10
1
2
3
45
Figure 3.1: 2D Euler equations. Case (1). The evolution of the
average residue. The resultsof RKDG methods with multi-resolution
WENO limiters. From left to right and top tobottom: second-order,
third-order, fourth-order, and fifth-order methods. Different
numbersindicate different mesh levels from 20×20 to 60×60
cells.
13
-
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5-16
-14
-12
-10
-8
-6
-4
-2
12
34
5
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5-16
-14
-12
-10
-8
-6
-4
-2
1
234
5
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5
-12
-10
-8
-6
1
2
34
5
Time
Lo
g1
0(R
esA
)
0 1 2 3 4 5
-12
-11
-10
-9
-8
1
2
3
4
5
Figure 3.2: 2D Euler equations. Case (2). The evolution of the
average residue. The resultsof RKDG methods with multi-resolution
WENO limiters. From left to right and top tobottom: second-order,
third-order, fourth-order, and fifth-order methods. Different
numbersindicate different mesh levels from 20×20 to 60×60
cells.
14
-
Table 3.2: 2D Euler equations. Case (2). RKDG methods with
multi-resolution WENOlimiters. Steady state. L1 and L∞ errors.
Second-order method Third-order methodGrid cells L1 error order
L∞ error order L1 error order L∞ error order20×20 4.60E-4 3.62E-3
2.43E-5 1.82E-430×30 1.91E-4 2.17 1.61E-3 1.99 7.32E-6 2.96 5.54E-5
2.9440×40 1.04E-4 2.09 9.10E-4 2.00 3.11E-6 2.98 2.36E-5 2.9650×50
6.61E-5 2.05 5.83E-4 2.00 1.59E-6 2.98 1.21E-5 2.9860×60 4.56E-5
2.03 4.05E-4 2.00 9.27E-7 2.99 7.04E-6 2.99
Fourth-order method Fifth-order methodGrid cells L1 error order
L∞ error order L1 error order L∞ error order20×20 4.51E-7 4.14E-6
1.27E-8 9.31E-830×30 8.77E-8 4.04 8.18E-7 4.00 1.64E-9 5.06 1.27E-8
4.9040×40 2.75E-8 4.03 2.59E-7 4.00 3.85E-10 5.04 3.08E-9 4.9550×50
1.12E-8 4.02 1.06E-7 4.00 1.25E-10 5.03 1.01E-9 4.9760×60 5.40E-9
4.01 5.12E-8 4.00 5.02E-11 5.02 4.10E-10 4.98
Initially, we set the solution in the entire domain to be that
at the left boundary. We show the
density contours with 15 equally spaced contour lines from 1.10
to 2.58 when steady states
are reached for different orders of RKDG methods with
multi-resolution WENO limiters
in Figure 3.3. The troubled cells identified at the final time
step are shown in Figure
3.4. We can clearly observe that the fifth-order RKDG method
with the associated multi-
resolution WENO limiter gives better resolution than that of the
lower order RKDGmethods,
especially for obtaining sharp shock transitions. The
convergence history of the residue (3.1)
as a function of time is shown in Figure 3.5. It can be observed
that the average residue of
second-order, third-order, fourth-order, and fifth-order
RKDGmethods with multi-resolution
WENO limiters can settle down to a value around 10−11.5, close
to machine zero.
Example 3.3. This problem is a supersonic flow past a plate with
an attack angle of α = 10◦.
The free stream Mach number is M∞ = 3. The ideal gas goes from
the left toward the plate.
The initial conditions are p = 1γM2∞
, ρ = 1, µ = cos(α), and ν = sin(α). The computational
field is [0, 10]× [−5, 5]. The plate is set at x ∈ [1, 2] with y
= 0. The slip boundary condition
is imposed on the plate. The physical values of the inflow and
outflow boundary conditions
15
-
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
Figure 3.3: The shock reflection problem. 15 equally spaced
density contours from 1.10 to2.58. The results of RKDG methods with
multi-resolution WENO limiters. From top tobottom: second-order,
third-order, fourth-order, and fifth-order methods. 120× 30
cells.
16
-
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
X
Y
0 1 2 3 40
0.5
1
Figure 3.4: The shock reflection problem. Troubled cells.
Squares denote cells which areidentified as troubled cells subject
to multi-resolution WENO limiting procedures at thelast time step.
From top to bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 120× 30 cells.
17
-
Time
Lo
g1
0(R
esA
)
0 5 10 15-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15-14
-12
-10
-8
-6
-4
-2
0
Figure 3.5: The shock reflection problem. The evolution of the
average residue of RKDGmethods with multi-resolution WENO limiters.
From left to right and top to bottom: second-order, third-order,
fourth-order, and fifth-order methods. 120× 30 cells.
18
-
are applied in different directions. The results are shown when
the numerical solutions reach
their steady states. We show 30 equally spaced pressure contours
from 0.02 to 0.24 computed
by the different orders of RKDGmethods with multi-resolution
WENO limiters in Figure 3.6.
The troubled cells identified at the final time step are shown
in Figure 3.7. The convergence
history of the residue (3.1) is shown in Figure 3.8. More
noticeably, the average residue of
the different orders of RKDG methods with multi-resolution WENO
limiters can settle down
to a value around 10−13, close to machine zero. Although the
boundary is very far away from
the plate, the waves including the shocks and the rarefaction
waves propagate to the far field
boundaries. This usually causes difficulties for the residue of
high-order numerical schemes
to settle down to machine zero, while it does not seem to cause
much trouble for the different
orders of RKDG methods with multi-resolution WENO limiters.
Example 3.4. This problem is a supersonic flow past two plates
with an attack angle of
α = 10◦. The free stream Mach number is M∞ = 3. The ideal gas
goes from the left toward
two plates. The initial conditions are p = 1γM2∞
, ρ = 1, µ = cos(α), and ν = sin(α). The
computational field is [0, 10]× [−5, 5]. The two plates are set
at x ∈ [1, 2] with y = −2 and
x ∈ [1, 2] with y = 2. The slip boundary condition is imposed on
the plates. The physical
values of the inflow and outflow boundary conditions are applied
in different directions.
The results are shown when the numerical solutions reach their
steady states. We show
30 equally spaced pressure contours from 0.02 to 0.24 computed
by the different orders of
RKDG methods with multi-resolution WENO limiters in Figure 3.9.
The troubled cells
identified at the final time step are shown in Figure 3.10. The
convergence history of the
residue (3.1) is shown in Figure 3.11. More noticeably, the
average residue of the high-order
RKDG methods with multi-resolution WENO limiters can settle down
to a value around
10−13, close to machine zero. Although the boundary is very far
away from two plates, the
waves including the shocks, the rarefaction waves, and their
interactions propagate to the far
field top, bottom, and right boundaries, respectively. This
usually causes difficulties for the
residue of high-order RKDG methods from settling down close to
machine zero, while it does
19
-
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
Figure 3.6: A supersonic flow past a plate with an attack angle.
30 equally spaced pressurecontours from 0.02 to 0.24 of RKDG
methods with multi-resolution WENO limiters. Fromleft to right and
top to bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 100× 100 cells.
20
-
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure 3.7: A supersonic flow past a plate with an attack angle.
Squares denote cells whichare identified as troubled cells subject
to multi-resolution WENO limiting procedures at thelast time step.
From left to right and top to bottom: second-order, third-order,
fourth-order,and fifth-order methods. 100× 100 cells.
21
-
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.8: A supersonic flow past a plate with an attack angle.
The evolution of the averageresidue of RKDG methods with
multi-resolution WENO limiters. From left to right and topto
bottom: second-order, third-order, fourth-order, and fifth-order
methods. 100×100 cells.
22
-
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
Figure 3.9: A supersonic flow past two plates with an attack
angle. 30 equally spaced pressurecontours from 0.02 to 0.24 of RKDG
methods with multi-resolution WENO limiters. Fromleft to right and
top to bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 100× 100 cells.
not cause any difficulties for the different orders of RKDG
methods with multi-resolution
WENO limiters.
Example 3.5. This problem is a supersonic flow past three plates
with an attack angle
of α = 10◦. The free stream Mach number is M∞ = 3. The ideal gas
goes from the left
toward the plates. The initial conditions are set as p =
1γM2∞
, ρ = 1, µ = cos(α), and
ν = sin(α). The computational field is [0, 10] × [−5, 5]. Three
plates are set at x ∈ [1, 2]
with y = −2, x ∈ [1, 2] with y = 0, and x ∈ [1, 2] with y = 2.
The slip boundary condition is
imposed on three plates. The physical values of the inflow and
outflow boundary conditions
23
-
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure 3.10: A supersonic flow past two plates with an attack
angle. Squares denote cellswhich are identified as troubled cells
subject to multi-resolution WENO limiting proceduresat the last
time step. From left to right and top to bottom: second-order,
third-order,fourth-order, and fifth-order methods. 100× 100
cells.
24
-
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.11: A supersonic flow past two plates with an attack
angle. The evolution ofthe average residue of RKDG methods with
multi-resolution WENO limiters. From left toright and top to
bottom: second-order, third-order, fourth-order, and fifth-order
methods.100× 100 cells.
25
-
are applied in different directions. The results are shown when
the numerical solutions
reach their steady states. We show 30 equally spaced pressure
contours from 0.02 to 0.24
computed by the different orders of RKDG methods with
multi-resolution WENO limiters in
Figure 3.12. The troubled cells identified at the final time
step are shown in Figure 3.13. The
convergence history of the residue (3.1) is shown in Figure
3.14. More noticeably, the average
residue of the second-order, third-order, fourth-order, and
fifth-order RKDG methods with
associated multi-resolution WENO limiters can settle down to a
tiny value around 10−13,
close to machine zero. Although the boundary is very far away
from the three plates, the
shock waves, the rarefaction waves, and their interaction waves
propagate to the far field
boundaries. It often causes the residue of high-order RKDG
methods with WENO limiters
from settling down to machine zero. But it does not seem to
cause much trouble for the
different orders of RKDG methods with multi-resolution WENO
limiters specified in this
paper.
Example 3.6. This problem is a supersonic flow past a long plate
with an attack angle of
α = 10◦. The free stream Mach number is M∞ = 3. The ideal gas
goes from the left toward
the long plate. The initial condition is set as p = 1γM2∞
, ρ = 1, µ = cos(α), and ν = sin(α).
The computational field is [0, 7]×[−5, 5]. The long plate region
is set as x ∈ [2, 7] with y = 0.
The slip boundary condition is imposed on the long plate. The
physical values of the inflow
and outflow boundary conditions are applied at the outer
boundaries. The results are shown
when the numerical solutions have settled down to their steady
states. We show 30 equally
spaced pressure contours from 0.031 to 0.161 computed by the
different orders of RKDG
methods with multi-resolution WENO limiters in Figure 3.15. The
troubled cells identified
at the final time step are shown in Figure 3.16. The convergence
history of the residue (3.1)
is shown in Figure 3.17. We can find that the average residue of
the different orders of RKDG
methods with multi-resolution WENO limiters settles down to a
value around 10−12.5, close
to machine zero. In this case, the shocks and the rarefaction
waves pass through the right
boundary. This is usually one reason that residue for high-order
schemes has difficulty from
26
-
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
X
Y
0 2 4 6 8 10
-4
-2
0
2
4
Figure 3.12: A supersonic flow past three plates with an attack
angle. 30 equally spacedpressure contours from 0.02 to 0.24 of RKDG
methods with multi-resolution WENO limiters.From left to right and
top to bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 100× 100 cells.
27
-
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
X
Y
0 2 4 6 8 10-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure 3.13: A supersonic flow past three plates with an attack
angle. Squares denote cellswhich are identified as troubled cells
subject to multi-resolution WENO limiting proceduresat the last
time step. From left to right and top to bottom: second-order,
third-order,fourth-order, and fifth-order methods. 100× 100
cells.
28
-
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 20 40 60 80
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.14: A supersonic flow past three plates with an attack
angle. The evolution ofthe average residue of RKDG methods with
multi-resolution WENO limiters. From left toright and top to
bottom: second-order, third-order, fourth-order, and fifth-order
methods.100× 100 cells.
29
-
settling down to machine zero, but it does not seem to affect
the high-order RKDG methods
with multi-resolution WENO limiters in this paper so much.
Example 3.7. This problem is a supersonic flow past two long
plates with an attack angle
of α = 10◦. The free stream Mach number is M∞ = 3. The ideal gas
goes from the left
toward two long plates. The initial condition is set as p =
1γM2∞
, ρ = 1, µ = cos(α), and
ν = sin(α). The computational field is [0, 7]× [−5, 5]. Two long
plates are set at x ∈ [2, 7]
with y = −2 and x ∈ [2, 7] with y = 2. The slip boundary
condition is imposed on two long
plates. The physical values of the inflow and outflow boundary
conditions are applied at the
left, right, bottom, and top boundaries. The results are shown
when the numerical solutions
have settled down to their steady states. We show 30 equally
spaced pressure contours from
0.031 to 0.161 computed by the different orders of RKDG methods
with multi-resolution
WENO limiters in Figure 3.18. The troubled cells identified at
the final time step are shown
in Figure 3.19. The convergence history of the residue (3.1) is
shown in Figure 3.20. We
can find that the average residue of the different orders of
RKDG methods with multi-
resolution WENO limiters settles down to a value around 10−12.5,
close to machine zero. In
this case, the shocks, the rarefaction waves, and their
interactions all pass through the right
boundary. It is one of the reasons that residues for many
high-order schemes do not converge
to machine zero, however this does not seem to be the case for
this second-order, third-order,
fourth-order, and fifth-order RKDG methods with new
multi-resolution WENO limiters.
Example 3.8. This problem is a supersonic flow past three long
plates with an attack angle
of α = 10◦. The free stream Mach number is M∞ = 3. The ideal gas
goes from the left
toward three long plates. The initial condition is set as p =
1γM2∞
, ρ = 1, µ = cos(α), and
ν = sin(α). The computational field is [0, 5]× [−5, 5]. Three
long plates are set at x ∈ [2, 5]
with y = −2, x ∈ [2, 5] with y = 0, and x ∈ [2, 5] with y = 2.
The slip boundary condition
is imposed on three long plates. The physical values of the
inflow and outflow boundary
conditions are applied at the left, right, bottom, and top
boundaries. The results are shown
30
-
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
Figure 3.15: A supersonic flow past a long plate problem. 30
equally spaced pressure contoursfrom 0.031 to 0.161 of RKDG methods
with multi-resolution WENO limiters. From left toright and top to
bottom: second-order, third-order, fourth-order, and fifth-order
methods.140× 200 cells.
31
-
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
Figure 3.16: A supersonic flow past a long plate problem.
Squares denote cells which areidentified as troubled cells subject
to multi-resolution WENO limiting procedures at the lasttime step.
From left to right and top to bottom: second-order, third-order,
fourth-order,and fifth-order methods. 140× 200 cells.
32
-
Time
Lo
g1
0(R
esA
)
0 10 20 30 40 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40 50
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.17: A supersonic flow past a long plate problem. The
evolution of the averageresidue of RKDG methods with
multi-resolution WENO limiters. From left to right and topto
bottom: second-order, third-order, fourth-order, and fifth-order
methods. 140×200 cells.
33
-
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
X
Y
0 2 4 6
-4
-2
0
2
4
Figure 3.18: A supersonic flow past two long plates problem. 30
equally spaced pressurecontours from 0.031 to 0.161 of RKDG methods
with multi-resolution WENO limiters. Fromleft to right and top to
bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 140× 200 cells.
34
-
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
X
Y
0 1 2 3 4 5 6 7
-4
-2
0
2
4
Figure 3.19: A supersonic flow past two long plates problem.
Squares denote cells which areidentified as troubled cells subject
to multi-resolution WENO limiting procedures at the lasttime step.
From left to right and top to bottom: second-order, third-order,
fourth-order,and fifth-order methods. 140× 200 cells.
35
-
Time
Lo
g1
0(R
esA
)
0 10 20 30 40
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 10 20 30 40
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.20: A supersonic flow past two long plates problem. The
evolution of the averageresidue of RKDG methods with
multi-resolution WENO limiters. From left to right and topto
bottom: second-order, third-order, fourth-order, and fifth-order
methods. 140×200 cells.
36
-
when the numerical solutions have settled down to their steady
states. We show 30 equally
spaced pressure contours from 0.031 to 0.161 computed by the
different orders of RKDG
methods with multi-resolution WENO limiters in Figure 3.21. The
troubled cells identified
at the final time step are shown in Figure 3.22. The convergence
history of the residue
(3.1) is shown in Figure 3.23. We can find that the average
residue of high-order RKDG
methods with multi-resolution WENO limiters settles down to a
value around 10−12.5, close
to machine zero. In this case, the shocks, the rarefaction
waves, and their interactions all
pass through the right boundary. It is one of the reasons that
residues for many high-order
schemes such as other high-order RKDG methods with WENO/HWENO
limiters do not
converge to machine zero, however this does not seem to be the
case for the RKDG methods
with multi-resolution WENO limiters in this paper.
4 Concluding remarks
In this paper, we design a new troubled cell indicator and adopt
our high-order finite vol-
ume multi-resolution WENO schemes [49] to serve as limiters for
high-order RKDG methods
to solve two-dimensional steady-state problems on structured
meshes. The general frame-
work of such multi-resolution WENO limiters for high-order RKDG
methods is to first design
a new methodology to detect troubled cells subject to the
multi-resolution WENO limiting
procedure, then to construct a sequence of hierarchical L2
projection polynomial solutions of
the DG methods completely restricted to the troubled cell itself
in a WENO fashion. To the
best of our knowledge, it is the first time that numerical
residue for second-order, third-order,
fourth-order, and fifth-order RKDGmethods with multi-resolution
WENO limiters can settle
down close to machine zero for benchmark steady-state problems,
including some problems
containing strong shocks, contact discontinuities, rarefaction
waves, their interactions, and
associated compound sophisticated waves passing through
boundaries. The results in this
paper indicate that these new high-order RKDG methods with
multi-resolution WENO lim-
iters have a good potential in computing the steady-state
problems, than other WENO
37
-
X
Y
0 2 4
-4
-2
0
2
4
X
Y
0 2 4
-4
-2
0
2
4
X
Y
0 2 4
-4
-2
0
2
4
X
Y
0 2 4
-4
-2
0
2
4
Figure 3.21: A supersonic flow past three long plates problem.
30 equally spaced pressurecontours from 0.031 to 0.161 of RKDG
methods with multi-resolution WENO limiters. Fromleft to right and
top to bottom: second-order, third-order, fourth-order, and
fifth-ordermethods. 100× 200 cells.
38
-
X
Y
0 1 2 3 4 5
-4
-2
0
2
4
X
Y
0 1 2 3 4 5
-4
-2
0
2
4
X
Y
0 1 2 3 4 5
-4
-2
0
2
4
X
Y
0 1 2 3 4 5
-4
-2
0
2
4
Figure 3.22: A supersonic flow past three long plates problem.
Squares denote cells whichare identified as troubled cells subject
to multi-resolution WENO limiting procedures at thelast time step.
From left to right and top to bottom: second-order, third-order,
fourth-order,and fifth-order methods. 100× 200 cells.
39
-
Time
Lo
g1
0(R
esA
)
0 5 10 15 20 25 30 35 40 45 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15 20 25 30 35 40 45 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15 20 25 30 35 40 45 50
-14
-12
-10
-8
-6
-4
-2
0
Time
Lo
g1
0(R
esA
)
0 5 10 15 20 25 30 35 40 45 50
-14
-12
-10
-8
-6
-4
-2
0
Figure 3.23: A supersonic flow past three long plates problem.
The evolution of the averageresidue of RKDG methods with
multi-resolution WENO limiters. From left to right and topto
bottom: second-order, third-order, fourth-order, and fifth-order
methods. 100×200 cells.
40
-
type limiters for the RKDG methods together with some classical
troubled cell indicators
[7, 8, 9, 10, 11, 28, 36, 37, 48].
The framework of this new type of multi-resolution WENO limiters
for arbitrary high-
order RKDG methods would be particularly efficient and simple
for solving steady-state
problems on unstructured meshes (such as triangular meshes or
tetrahedral meshes), and
the study of which is our ongoing work.
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46
