1. Hybrid WENO-FD and RKDG Method for Hyperbolic Conservation Laws
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

2. Outline Introduction RKDG and WENO-FD methods
BeiHang
University
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work

3. 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

4. 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

5. Introduction WENO-FD vs WENO-FV vs RKDG*: 2D Scalar 3D Euler 4*2 4/2
method
Surface GPs
Volume GPs
Reconst
Surface flux
Surface Integral
Volume Integral
2D Scalar
3D Euler
3rd-WENO-FD
4*2
4/2 10 75
5th-WENO-FD
4*3 14
100
3rd-WENO-FV
2*4
8*4
4*2/2 38
1035
5th-WENO-FV
3*4
12*9
3*4/2
116
7440
3rd-DG 9
6*4/2 5 44
735
5th-DG
5*4 25
5*4/2
15*4/2 99
2425
Bold: computational cost;
* : without limiter

6. Introduction
Hybrid techniques might be the way for solving 3D high Reynolds number compressible flow
Hybrid Mesh : AMR, DDM
Hybrid Methodology

7. Introduction Multi-domain methods Hybrid methods
Hybrid finite compact-WENO scheme
Multi-domain hybrid spectral-WENO methods
Etc.
Patched grids
Overlapping grids

8. Outline Introduction RKDG and WENO-FD methods
BeiHang
University
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work

9. RKDG methods Two dimensional hyperbolic conservation laws:
Spatial discretization:
The solution and test function space:
DG adopts a series of local basis over target cell:
The numerical solution can be written as:

10. RKDG methods Time discretization:
Multiply test functions and integrate over target cell:
where
On cell boundaries, the numerical solution is discontinuous, a numerical flux based on Riemann solution is used to replace the original flux:
Time discretization:
third-order Runge-Kutta method

11. WENO methods WENO-FD WENO-FV
(finite difference based WENO)
WENO-FV
(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!

12. WENO-FD schemes Two dimensional hyperbolic conservation laws:
Spatial discretization:
For a WENO-FD scheme, uniform grid is required and solve directly using
a conservative approximation to the space derivative:
The numerical fluxes are obtained by one dimensional WENO-FD
approximation procedure.

13. 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:

14. 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:
The nonlinear weights are given by:
The numerical flux for 5th order WENO-FD:

15. Summary RKDG WENO-FD Advantage Well in handling complex geometries
Highly efficient in structured grid
Weakness
Expensive in computational costs and storage requirements
Only in uniform mesh and hard in handling complex geometry

16. Outline Introduction RKDG and WENO-FD methods
BeiHang
University
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work

17. Multidomain hybrid 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

18. RKDG+WENO-FD method on structured meshes
RKDG+WENO-FD method for one
dimensional conservation laws
In WENO-FD domain
In RKDG domain
Conservative coupling method:
Non-conservative coupling method:
5th order WENO-FD+3rd order RKDG → Non-conservative Coupling

19. RKDG+WENO-FD method on treatment of shock wave
1D non-conservative RKDG+WENO-FD method:
In WENO-FD domain
In RKDG domain
If one of the cells j and j+1 is polluted, let

20. RKDG+WENO-FD method: theoretical results
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):
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.

21. 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

22. 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
at target cell
Second, we construct degrees of freedom of RKDG
at the target cell with a local orthogonal basis
At last, we get Gauss quadrature point values and
form the interface flux for RKDG

23. RKDG+WENO-FD method on hybrid meshes
Indicator of polluted cell:
We define
and
A TVD(TVB) smooth indicator is applied at the coupling interface to indicate possible discontinuities:
where m(a1,a2,a3) is TVD(TVB) minmod function.

24. RKDG+WENO-FD method on hybrid meshes
Hybrid RKDG+WENO-FD approach
Initiate WENO and RKDG data;
Construct Lagrange interpolation at the interface and use coupling
interface indicator;
Choose the coupling scheme at the interface:
(1)If the stencil is polluted, choose conservative coupling at the
interface(WENO-FD flux is used);
(2)if solution in the stencil is smooth enough, choose non-conservative
coupling.
Space discretization RKDG and WENO-FD domain;
Time discretization.

25. Outline Introduction RKDG and WENO-FD methods
BeiHang
University
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work

26. 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.
Hybrid mesh
(h=1/20)

27. 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.

28. Numerical results: 1D Euler systems
Example 3: Sod’s Shock Tube Problem
Artificial boundary at x=-0.5 & 0.5, t=0.4

29. 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

30. 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.

31. 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
(b)Hybrid RKDG+WENO-FD, h=1/120, CPU times: s
(c) RKDG, h=1/120, CPU times: s
(d) WENO-FD, h=1/120, CPU times: 607.8s
FIG. Double mach reflection problem

32. 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.
(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: s.
(a)Interaction of isentropic vortex and weak shock wave, sample mesh, mesh
size ℎ = 1/20.
(d) 5𝑡ℎ-WENO-FD scheme, density 30
contours from 1.0 to 1.24, mesh size
h=1/100, t=0.4, CPU time: 37.88s.
(c) 3𝑟𝑑-RKDG method, density 30 contours
from 1.0 to 1.24, mesh size h=1/100, t=0.4, CPU time: s.

33. 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.
FIG. Flow through a channel with a smooth bump, sample mesh, mesh size ℎ = 1/20.

34. 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.

35. 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.

36. 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: s.
(b) 3𝑟𝑑- RKDG method, pressure 25 contours from 1.0 to 20.0, mesh size h=1/100,t=0.5, CPU time: s.

37. 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.

38. Numerical results: 2D Euler systems
Example 11: Subsonic flow past NACA0012 airfoil

39. Outline Introduction RKDG and WENO-FD methods
BeiHang
University
Outline
Introduction
RKDG and WENO-FD methods
Hybrid RKDG+WENO-FD method
Numerical results
Conclusions and future work

40. 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.

41. Future work Accelerate convergence for steady flow
Adopt local mesh refinement and cut-cell approach
Extend to two dimensional N-S equations

42. BeiHang
University
Thank YOU!
BeiHang University