hybrid weno-fd and rkdg method for hyperbolic conservation laws tiegang liu school of mathematics...
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Hybrid WENO-FD and RKDG Method for Hyperbolic Conservation Laws
Tiegang Liu
School of Mathematics and Systems Science BeiHang University
10-14 Sept, 2013
BeiHangUniversity
Joint work with Jian Cheng
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHangUniversity
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◦ 1 order faster 3rd order flow solver
◦ 1 order faster optimization solver under nonlinear N-S equation constrains
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*:
Bold: computational cost; * : without limiter
method Surface GPs Volume GPs Reconst Surface flux Surface Integral
Volume Integral
2D Scalar
3D Euler
3rd-WENO-
FD0 0 4*2 4/2 0 0 10 75
5th-WENO-
FD0 0 4*3 4/2 0 0 14 100
3rd-WENO-
FV2*4 0 8*4 4*2/2 4/2 0 38 1035
5th-WENO-
FV3*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
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
• Hybrid methods– Hybrid finite compact-WENO scheme– Multi-domain hybrid spectral-WENO methods– Etc.
Patched grids Overlapping grids
Introduction
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work
BeiHangUniversity
RKDG methods
Two dimensional hyperbolic conservation laws:
0
( ) ( ) 0 in (0, )
( , ,0) ( , )
t x yu f u g u T
u x y u x y
The solution and test function space:
{ ( , ) : ( , ) | ( )}j
K kh jV v x y v x y P
DG adopts a series of local basis over target cell:
( ){ ( , ), 0,1,..., ; ( 1)( 2) / 2 1}lv x y l K K k k
The numerical solution can be written as:
( )( , , ) ( ) ( , )h ll
l
u x y t u t v x y
Spatial discretization:
RKDG methods
Multiply test functions and integrate over target cell:
( ) ( )
( ) ( )
( , ) ( ( ), ( )) ( , )
( ( ) ( , ) ( ) ( , )) 0
j j
j
h l h h T l
h l h l
du v x y dxdy f u g u nv x y ds
dt
f u v x y g u v x y dxdyx y
0, ...,l k
where ( , )x yn n n
On cell boundaries, the numerical solution is discontinuous, a numerical flux based on Riemann solution is used to replace the original flux:
,( ( ), ( )) ( , )j
h h Tnf u g u n h u u
Time discretization:
third-order Runge-Kutta method
WENO-FD(finite difference based WENO)
WENO methods
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-FV(finite volume based WENO)
WENO-FD schemes
Two dimensional hyperbolic conservation laws:
0
( ) ( ) 0 in (0, )
( , ,0) ( , )
t x yu f u g u T
u x y u x y
For a WENO-FD scheme, uniform grid is required and solve directly using a conservative approximation to the space derivative:
,1 1 1 1
, , , ,2 2 2 2
1 1ˆ ˆ ˆ ˆ( ) ( ) 0i j
i j i j i j i j
duf f g g
dt x y
The numerical fluxes are obtained by one dimensional WENO-FD approximation procedure.
1 1, ,
2 2
ˆ ˆ, ,i j i jf g
Spatial discretization:
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.
The numerical flux for 5th order WENO-FD:
0 0 0 0 31 1 2 2 1 3
2
1 1 1 1 31 1 2 2 1 3
2
2 2 2 2 31 1 2 2 1 3
2
( )
( )
( )
j j jj
j j jj
j j jj
q a f a f a f O x
q a f a f a f O x
q a f a f a f O x
0 1 21 0 1 1 1 2 1
2 2 2 2
ˆj j j j
f d q d q d q
WENO-FD schemes
Classical WENO schemes use the smooth indicator(Jiang and Shu JCP,1996) of each stencil as follows:
The nonlinear weights are given by:
One dimensional WENO-FD procedure: (5th-order WENO-FD)
1/2
1/2
12 1 2
1
( )( )
j
j
l kr x lk lx
l
q xx dx
x
1 2
0
0,1,..., 1( )
k kk kr
kss
dw k r
The numerical flux for 5th order WENO-FD:
0 1 21 0 1 1 1 2 1
2 2 2 2
ˆj j j j
f w q w q w q
Summary
RKDG WENO-FD
Advantage Well in handling complex geometries
Highly efficient in structured grid
WeaknessExpensive 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
BeiHangUniversity
Multidomain hybrid RKDG+WENO-FD method
RKDG+WENO-FD method
Couple RKDG and WENO-FD based on domain decompositionCombine 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 RKDG domain
In WENO-FD domain
( ) ( )1 1
2 2
1 ˆ ˆ( ) 1,...,WENO WENOk
k kk
duf f k j
dt x
( ) ( )1 1
2 2
1 ˆ ˆ( ) 1,...,DG DGk
k kk
duf f k j N
dt x
Conservative coupling method:
Non-conservative coupling method:
( ) ( )1 1
2 2
ˆ ˆWENO DG
j jf f
( ) ( )1 1
2 2
ˆ ˆWENO DG
j jf f
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 RKDG domain
In WENO-FD domain
( ) ( )1 1
2 2
1 ˆ ˆ( ) 1,...,WENO WENOk
k kk
duf f k j
dt x
( ) ( )1 1
2 2
1 ˆ ˆ( ) 1,...,DG DGk
k kk
duf f k j N
dt x
If one of the cells j and j+1 is polluted, let( ) ( )
1 1
2 2
ˆ ˆDG WENO
j jf f
RKDG+WENO-FD method: theoretical results
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 qth-order WENO-FD can preserve rth-order (r=min(p,q)) accuracy in smooth region.
Conservation error:
The conservative error of non-conservative multi-domain hybrid method of pth-order RKDG and qth-order WENO-FD is of 3rd-order accuracy.
We consider a general form of the hybrid RKDG+WENO-FD method which a pth-order DG method couples with a qth-order WENO-FD scheme (SISC, 2013):
1| |n nj j j j
j j
CE x u x u
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, wefollow these three steps:
First, we construct a high order polynomial at target cell
( , )p x y,i jI
Second, we construct degrees of freedom of RKDG at the target cell with a local orthogonal basis
,,
,,
( ) ( ), ( ) 2
1( , ) ( , )
( ( , )) i ji j
i ji j
l li j Il I
II
u p x y v x y dxdyv x y dxdy
At last, we get Gauss quadrature point values and form the interface flux for RKDG
( )ˆ ˆ ( , )RKDG LFIf f u u
RKDG+WENO-FD method on hybrid meshes
Indicator of polluted cell:
,i jI 1,i jI eI
( )rgu
We define ( )
,r
i j gu u u
1, ,i j i ju u u ,i j eu u u
and
A TVD(TVB) smooth indicator is applied at the coupling interface to indicate possible discontinuities:
(mod)( , , )u 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
BeiHangUniversity
Numerical results: Accuracy tests
Example 1: 2D linear scalar conservation law
0 ( , ) (0,1) (0,1)
( ,0) sin(2 )sin(2 )
t x yu u u x y
u x x y
Hybrid mesh
(h=1/20)
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.
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.
2 21 1( ) ( ) 0 ( , ) ( 1,1) ( 1,1)2 2
( , ,0) 0.5sin( ( )) 0.25
t x yu u u x y
u x y x y
0.5 /t
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
(a)WENO-FD scheme (b) RKDG-WENO-FD hybrid method
Artificial boundary at x=0.25 & 0.75, t=0.038
Example 4: Two Interacting Blast Waves
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.
1.5 /t
Numerical results: 2D Euler systems
Example 6: Double Mach Reflection
(a)Hybrid mesh h=1/20, interface y=0.2 (b)Hybrid RKDG+WENO-FD, h=1/120, CPU times: 39894.4s
(c) RKDG, h=1/120, CPU times: 92743.4s (d) WENO-FD, h=1/120, CPU times: 607.8s
FIG. Double mach reflection problem
This is a standard test case for high resolution schemes which a mach 10 shock initially makes a 60o angle with a reflecting wall.
Numerical results: 2D Euler systems
Example 7: Interaction of isentropic vortex and weak shock wave
(a)Interaction of isentropic vortex and weak shock wave, sample mesh, meshsize = 1/20.ℎ
(b) 3 -hybrid 𝑟𝑑RKDG+WENO-FDmethod, density 30 contours from 1.0to 1.24, mesh size h=1/100, t=0.4,CPU time: 2148.7s.
(c) 3 -RKDG 𝑟𝑑method, density 30 contoursfrom 1.0 to 1.24, mesh size h=1/100, t=0.4, CPU time: 4105.9s.
(d) 5 -WENO-𝑡ℎFD scheme, density 30contours from 1.0 to 1.24, mesh sizeh=1/100, t=0.4, CPU time: 37.88s.
This problem describes the interaction between a moving vortex and a stationary shock wave.
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
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.
2250.0625 xy e
FIG. Flow through a channel with a smooth bump, sample mesh, mesh size = 1/20.ℎ
Numerical results: 2D Euler systems
Example 8: Flow through a channel with a smooth bump
FIG. Subsonic flow, 3 -hybrid RKDG+WENO-FD method, Mach number 𝑟𝑑15 contours from 0.44 to 0.74, mesh size h=1/20.
FIG. Supersonic flow,3 -hybrid RKDG+WENO-FD method, density 25 𝑟𝑑contours 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 = −1.5 to = 1.5 𝑥 𝑥and height for = −1.0 to = 1.0 with a cylinder at the center. The diameter of 𝑦 𝑦the 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 = −1.0.𝑥
FIG. Comparison of sample Mesh. Left for RKDG; Right for hybrid RKDG+WENOFD, mesh size = 1/20.ℎ
Numerical results: 2D Euler systems
Example 9: Incident shock past a cylinder
(a) 3 - 𝑟𝑑 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) 3 - RKDG method, pressure 25 𝑟𝑑contours 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 3 -hybrid RKDG+WENO-FD method, 𝑟𝑑20 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
BeiHangUniversity
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
