#### Transcript Tiegang Liu, Beijing University of Aeronautics and Astronautics. The

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 ui1, 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 [email protected] BeiHang University