BeiHang
University
Hybrid WENO-FD and RKDG Method for Hyperbolic
Conservation Laws
Tiegang Liu
School of Mathematics and Systems Science
BeiHang University
10-14 Sept, 2013
Joint work with Jian Cheng
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHang
University
Introduction



Adjoint method: requiring steady viscous flow field computation for 5x102~103
intermediate shapes of aircrafts (convergence error of 10-6)
5~10hours parallel computation for a steady viscous flow field passing over a
whole aircraft by usual 2nd order high resolution methods with multigrid
technique
Totally 3monthes ~ half year computation for completing one round of design
Introduction


Strategy
Currently available 3rd order high resolution
methods
◦
◦
◦
◦
◦
DG: RKDG, HDG, …
Finite difference WENO (WENO-FD)
Finite volume WENO (WENO-FV)
Compact Schemes
Spectrum Methods
Introduction
WENO-FD vs WENO-FV vs RKDG*:
method
Surface GPs
Volume GPs
Reconst
Surface flux
Surface Integral
2D
3D Euler
Scalar
Volume
Integral
3rdWENO-FD
0
0
4*2
4/2
0
0
10
75
5thWENO-FD
0
0
4*3
4/2
0
0
14
100
3rdWENO-FV
2*4
0
8*4
4*2/2
4/2
0
38
1035
5thWENO-FV
3*4
0
12*9
3*4/2
4/2
0
116
7440
3rd-DG
3*4
9
0
3*4/2
6*4/2
5
44
735
5th-DG
5*4
25
0
5*4/2
15*4/2
14
99
2425
Bold: computational cost;
* : without limiter
Introduction
• Hybrid techniques might be the way for solving 3D
high Reynolds number compressible flow
– Hybrid Mesh : AMR, DDM
– Hybrid Methodology
Introduction
• Multi-domain methods
Patched grids
Overlapping grids
• Hybrid methods
– Hybrid finite compact-WENO scheme
– Multi-domain hybrid spectral-WENO methods
– Etc.
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHang
University
RKDG methods
Two dimensional hyperbolic conservation laws:
ut  f (u ) x  g (u ) y  0

u ( x, y, 0)  u0 ( x, y )
in   (0, T )
Spatial discretization:
The solution and test function space:
VhK  {v( x, y) : v( x, y) | j  P k ( j )}
DG adopts a series of local basis over target cell:
{v(l ) ( x, y), l  0,1,..., K ; K  (k 1)(k  2) / 2 1}
The numerical solution can be written as:
u h ( x, y, t )   ul (t )v(l ) ( x, y)
l
RKDG methods
Multiply test functions and integrate over target cell:
d
h (l )
h
h T
(l )
u
v
(
x
,
y
)
dxdy

(
f
(
u
),
g
(
u
))

nv
( x, y )ds




j
j
dt


  ( f (u h ) v (l ) ( x, y )  g (u h ) v (l ) ( x, y ))dxdy  0
j
x
y
l  0,..., k
where n  (nx , ny )
On cell boundaries, the numerical solution is discontinuous, a numerical flux
based on Riemann solution is used to replace the original flux:
( f (u h ), g (u h ))T  n  h j ,n (u  , u  )
Time discretization:
third-order Runge-Kutta method
WENO methods
 WENO-FD
 WENO-FV
(finite difference based WENO)
(finite volume based WENO)
Efficient for structured mesh
Not applicable for unstructured
mesh
Difficult in treatment of complex
boundaries
Easy in treatment of complex
boundaries
Costly and troublesome for
maintaining higher order for
unstructured mesh
WENO-FV has computational count 4 times (2D)/9
times (3D)larger than WENO-FD for 3rd order
accuracy!
WENO-FD schemes
Two dimensional hyperbolic conservation laws:
ut  f (u ) x  g (u ) y  0

u ( x, y, 0)  u0 ( x, y )
in   (0, T )
Spatial discretization:
For a WENO-FD scheme, uniform grid is required and solve directly using
a conservative approximation to the space derivative:
dui , j
dt

1 ˆ
1
( f 1  fˆ 1 ) 
( gˆ 1  gˆ 1 )  0
i

,
j
i

,
j
i , j
x
y i  2 , j
2
2
2
ˆ
The numerical fluxes fi  1 , j , gˆ i  1 , j , are obtained by one dimensional WENO-FD
2
approximation procedure.
2
WENO-FD schemes
One dimensional WENO-FD procedure: (5th-order WENO-FD)
WENO construct polynomial q (x) on each candidate stencil S0,S1,S2 and use the
convex combination of all candidate stencils to achieve high order accurate.
q0
 a10 f j 2  a20 f j 1  a30 f j  O(x3 )
q1
 a11 f j 2  a12 f j 1  a31 f j  O(x3 )
q2
 a12 f j 2  a22 f j 1  a32 f j  O(x3 )
1
j
2
1
j
2
1
j
2
The numerical flux for 5th order WENO-FD:
fˆ
1
j
2
 d0 q0 1  d1q1
j
2
1
j
2
 d2 q 2
j
1
2
WENO-FD schemes
One dimensional WENO-FD procedure: (5th-order WENO-FD)
Classical WENO schemes use the smooth indicator(Jiang and Shu JCP,1996)
of each stencil as follows:
r 1
k   
l 1
x j 1/2
x j 1/2
x
2l 1
 l q k ( x) 2
( l
) dx
x
The nonlinear weights are given by:
wk 
k

r 1
s 0
s
k 
dk
(   k )2
k  0,1,..., r  1
The numerical flux for 5th order WENO-FD:
fˆ
1
j
2
 w0q0 1  w1q1
j
2
1
j
2
 w2q2
j
1
2
Summary
Advantage
Weakness
RKDG
WENO-FD
Well in handling complex
geometries
Highly efficient in structured grid
Expensive in computational
costs and storage
requirements
Only in uniform mesh and hard in
handling complex geometry
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHang
University
Multidomain hybrid RKDG+WENO-FD method
RKDG+WENO-FD method
Couple RKDG and WENO-FD based on domain decomposition
Combine advantages of both RKDG and WENO-FD, 90-99%domain
in WENO-FD and 10-1%domain in RKDG
RKDG
WENO-FD
Hybrid mesh approach
Cut-cell approach
RKDG+WENO-FD method on structured meshes
RKDG+WENO-FD method for one
dimensional conservation laws
In WENO-FD domain
duk
1 ˆ (WENO ) ˆ (WENO )

(f 1
f 1 )
k

k
dt
xk
2
2
k  1,..., j
duk
1 ˆ ( DG ) ˆ ( DG )

(f 1  f 1 )
k
dt
xk k  2
2
k  j  1,..., N
In RKDG domain
Conservative coupling method:
Non-conservative coupling method:
)
ˆ ( DG )
fˆ (WENO

f
1
1
j
j
2
2
)
ˆ ( DG )
fˆ (WENO

f
1
1
j
2
j
2
5th order WENO-FD+3rd order RKDG → Non-conservative Coupling
RKDG+WENO-FD method on treatment of shock wave
1D non-conservative RKDG+WENO-FD
method:
In WENO-FD domain
duk
1 ˆ (WENO ) ˆ (WENO )

(f 1
f 1 )
k

k
dt
xk
2
2
k  1,..., j
duk
1 ˆ ( DG ) ˆ ( DG )

(f 1  f 1 )
k
dt
xk k  2
2
k  j  1,..., N
In RKDG domain
If one of the cells j and j+1 is polluted, let
)
ˆ (WENO)
fˆ ( DG

f
1
1
j
2
j
2
RKDG+WENO-FD method: theoretical results
We consider a general form of the hybrid RKDG+WENO-FD method which a pthorder DG method couples with a qth-order WENO-FD scheme (SISC, 2013):
Accuracy:
Conservative multi-domain hybrid method of pth-order RKDG and qth-order
WENO-FD is of 1st-order accuracy.
Non-conservative multi-domain hybrid method of pth-order RKDG and qthorder WENO-FD can preserve rth-order (r=min(p,q)) accuracy in smooth region.
Conservation error:
CE |  x j u nj 1  x j u nj |
j
j
The conservative error of non-conservative multi-domain hybrid method of
pth-order RKDG and qth-order WENO-FD is of 3rd-order accuracy.
RKDG+WENO-FD method on interface flux
Construct WENO-FD flux
When construct WENO-FD flux, RKDG can provide the central point values
for WENO construction.
FIG. RKDG provides point values for WENO-FD construction
RKDG+WENO-FD method on hybrid meshes
Construct RKDG flux
When construct RKDG flux, we use WENO point values to construct RKDG flux, we
follow these three steps:
First, we construct a high order polynomial p ( x, y )
at target cell I i , j
Second, we construct degrees of freedom of RKDG
at the target cell with a local orthogonal basis
ui(,lj) 
1

Ii , j
(vI(il,)j ( x, y)) 2 dxdy

Ii , j
p( x, y )vI(il,)j ( x, y )dxdy
At last, we get Gauss quadrature point values and
form the interface flux for RKDG
fˆI( RKDG)  fˆ LF (u , u )
RKDG+WENO-FD method on hybrid meshes
Indicator of polluted cell:
We define
u  u i , j  ug( r )
Ie
and
u  ui1, j  ui, j
 u  u i , j  u e
Ii, j
Ii 1, j
ug( r )
A TVD(TVB) smooth indicator is applied at the coupling interface to indicate
possible discontinuities:
u
(mod)
 m(u ,   u ,  u )
where m(a1,a2,a3) is TVD(TVB) minmod function.
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHang
University
Numerical results: Accuracy tests
Example 1: 2D linear scalar conservation law
We test the accuracy of the hybrid RKDG+WENO-FD method when applied to
solve a two dimensional linear scalar conservation law with exact boundary
condition couple interface at x=0.5, 0<y<1.
ut  ux  u y  0 ( x, y)  (0,1)  (0,1)

u ( x,0)  sin(2 x)sin(2 y)
Hybrid mesh
(h=1/20)
Numerical results: Accuracy tests
Example 2: 2D scalar Burgers’ equation
We test the accuracy of the hybrid RKDG+WENO-FD method when applied to
solve a two dimensional Burgers’ equation with exact boundary condition
couple interface at x=0, -1<y<1.
1 2
1 2

ut  ( u ) x  ( u ) y  0 ( x, y )  (1,1)  (1,1)
2
2

u ( x, y,0)  0.5sin( ( x  y ))  0.25
t  0.5 / 
Numerical results: 1D Euler systems
Example 3: Sod’s Shock Tube Problem
Artificial boundary at x=-0.5 & 0.5, t=0.4
Numerical results: 1D Euler systems
Example 4: Two Interacting Blast Waves
(a)WENO-FD scheme
(b) RKDG-WENO-FD hybrid method
Artificial boundary at x=0.25 & 0.75, t=0.038
Numerical results: 2D scalar conservation law
Example 5: 2D scalar Burgers’ equation
We test the accuracy of the hybrid RKDG+WENO-FD method when applied to
solve a two dimensional Burgers’ equation with exact boundary condition
couple interface at x=0, -1<y<1.
t  1.5 / 
Numerical results: 2D Euler systems
Example 6: Double Mach Reflection
This is a standard test case for high resolution schemes which a mach 10 shock
initially makes a 60o angle with a reflecting wall.
(a)Hybrid mesh h=1/20, interface y=0.2
(c) RKDG, h=1/120, CPU times: 92743.4s
(b)Hybrid RKDG+WENO-FD, h=1/120, CPU times: 39894.4s
(d) WENO-FD, h=1/120, CPU times: 607.8s
FIG. Double mach reflection problem
Numerical results: 2D Euler systems
Example 7: Interaction of isentropic vortex and weak shock wave
This problem describes the interaction between a moving vortex and a
stationary shock wave.
(a)Interaction of
isentropic vortex
and weak shock
wave, sample
mesh, mesh
size ℎ = 1/20.
(c) 3𝑟𝑑-RKDG
method, density
30 contours
from 1.0 to 1.24,
mesh size
h=1/100, t=0.4,
CPU time:
4105.9s.
(b) 3𝑟𝑑-hybrid
RKDG+WENO-FD
method, density 30
contours from 1.0
to 1.24, mesh size
h=1/100, t=0.4,
CPU time:
2148.7s.
(d) 5𝑡ℎ-WENOFD scheme,
density 30
contours from 1.0
to 1.24, mesh
size
h=1/100, t=0.4,
CPU time:
37.88s.
Numerical results: 2D Euler systems
Example 8: Flow through a channel with a smooth bump
The computational domain is bounded between x = -1.5 and x = 1.5, and between
the bump and y = 0.8. The bump is defined as
25 x2
y  0.0625e
We test two cases which is a subsonic flow with inflow Mach number is 0.5 with 0
angle of attack and a supersonic flow with Mach 2.0 with 0 angle of attack.
FIG. Flow through a channel with a smooth bump, sample mesh,
mesh size ℎ = 1/20.
Numerical results: 2D Euler systems
Example 8: Flow through a channel with a smooth bump
FIG. Subsonic flow, 3𝑟𝑑-hybrid RKDG+WENO-FD method, Mach number 15
contours from 0.44 to 0.74, mesh size h=1/20.
FIG. Supersonic flow,3𝑟𝑑-hybrid RKDG+WENO-FD method, density 25
contours from 0.55 to 1.95, mesh size h=1/50.
Numerical results: 2D Euler systems
Example 9: Incident shock past a cylinder
The computational domain is a rectangle with length from 𝑥 = −1.5 to 𝑥 = 1.5
and height for 𝑦 = −1.0 to 𝑦 = 1.0 with a cylinder at the center. The diameter of
the cylinder is 0.25 and its center is located at (0, 0). The incident shock wave
is at Mach number of 2.81 and the initial discontinuity is placed at 𝑥 = −1.0.
FIG. Comparison of sample Mesh. Left for RKDG; Right for
hybrid RKDG+WENOFD, mesh size ℎ = 1/20.
Numerical results: 2D Euler systems
Example 9: Incident shock past a cylinder
(a) 3𝑟𝑑- hybrid RKDG+WENO-FD
method, pressure 25 contours from
1.0 to 20.0, mesh size h=1/100,
t=0.5, CPU time: 7100.6s.
(b) 3𝑟𝑑- RKDG method, pressure 25
contours from 1.0 to 20.0, mesh size
h=1/100,t=0.5, CPU time: 41266.7s.
Numerical results: 2D Euler systems
Example 10: Supsonic flow past a tri-airfoil
This is a test of supersonic flow past three airfoils(NACA0012) with Mach number
1.2 and the attack angle 0.0∘. In the sample hybrid mesh for this test case,
unstructured meshes are applied in domain [−1.0, 3.0]×[−2.0, 2.0] around airfoils
and structured meshes used other computational domains.
FIG. Left Sample hybrid mesh, mesh size h=1/10. Right 3𝑟𝑑-hybrid RKDG+WENO-FD method,
20 density contours from 0.7 to 1.8, mesh size h=1/20.
Numerical results: 2D Euler systems
Example 11:
Subsonic flow past
NACA0012 airfoil
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHang
University
Conclusions
A relative simple approach is presented to combine a point-value based
WENO-FD scheme with an averaged-value based RKDG method to higher
order accuracy.
Special strategy is applied at coupling interface to preserve high order
accurate for smooth solution and avoid loss of conservation for
discontinuities.
Numerical results are demonstrated the flexibility of the hybrid
RKDG+WENO-FD method in handling complex geometries and the
capability of saving computational cost in comparison to the traditional
RKDG method.
Future work
Accelerate convergence for steady flow
Adopt local mesh refinement and cut-cell approach
Extend to two dimensional N-S equations
BeiHang
University
liutg@.buaa.edu.cn
BeiHang University