High-order Runge-Kutta discontinuous Galerkin methods with ... · to machine zero. Zhang et al....
Transcript of High-order Runge-Kutta discontinuous Galerkin methods with ... · to machine zero. Zhang et al....
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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.
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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
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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.
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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-
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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-
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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.
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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)
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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.
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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-
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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.
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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,∞) =
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-
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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