BeiHang University Multi-domain Hybrid RKDG and WENO-FD Method for Hyperbolic Conservation Laws Tiegang Liu School of Mathematics and Systems Science Beijing University of Aeronautics and Astronautics(Beihang University) 22-26 May, 2014 J i t Joint work with Jian Cheng, Kun Wang k ith Ji Ch K W
BeiHang Outline University Introduction Introduction RKDG and WENO FD methods RKDG and WENO-FD methods Hybrid RKDG+WENO FD method Hybrid RKDG+WENO-FD method Numerical results Numerical results Conclusions and future work Conclusions and future work
Introduction—3 rd order or higher is necessary 2 nd order methods do not Higher order efficient methods meet the requirements are demanding for 3D complex flow Simulations flow Simulations 2 nd order results o de esu s Complex flow requires Complex flow requires higher order methods Higher order enables lower mesh numbers 3 rd order results Order of Mesh number per unit method volume 2 nd 3 rd
Introduction—Cost & Efficiency Comparison of computational costs for Surfa popular higher order methods 3D Surf Vol Surf Volume Reco ce Quadrilater Quadrilater ace ace ume ume ace ace Integr Integr Sum Sum nst Integ al GPs GPs flux al ral Higher order 6 0 0 6 0 0 12 2 nd -JST FD methods 0 0 6*2 6 0 0 18 3 rd -WENO-FD 5 th WENO FD 0 0 0 0 6*3 6 3 6 6 0 0 0 0 24 24 5 th -WENO-FD 4*6 0 24*8 4*6 4*6 0 264 Good: Cheaper (Comparable to 2 nd 3 rd -WENO-FV 9*6 27 0 9*6 9*6 10 199 order FV method) 3 rd -DG Bad : Uniform mesh; not for complex geometry Higher order methods Grid No per unit Comput. Cost Comput. Cost FV/DG methods volume volume per grid cell per grid cell per unit volume per unit volume Good: unstructured mesh; complex 2 nd order FV geometry 3 rd order FD Bad : expensive (1 order higher than p ( g 2 nd methods) 3 rd order FV/DG A single higher order method does not work out well for 3D complex flow over complex geometry p p g y
Introduction—Hybrid technique • Multi ‐ domain methods Multi domain methods Patched grids Overlapping grids • Hybrid methods – Hybrid finite compact ‐ WENO scheme Hybrid finite compact WENO scheme – Multi ‐ domain hybrid spectral ‐ WENO methods – Etc Etc.
• Hybrid methods based on reconstruction – Balsara et al. [JCP, 2007], hybrid RKDG + HWENO schemes – Luo et al. [AIAA, 2010], Reconstructed DG Luo et al [AIAA 2010] Reconstructed DG – Dumbser et al. [JCP, 2008] one step finite volume +DG, PnPm – Zhang et al. [JCP, 2012], FV+DG g [ , ], • Hybrid methods based on domain decomposition – Costa et al. [JCP, 2007], hybrid spectral-WENO methods – Shahbazi et al. [JCP, 2007], Fourier-continuation/WENO – Zhu et al. [CiCP, 2011], hybrid finite difference and finite element time domain (FDTD/FETD) method (Maxwell equations) domain (FDTD/FETD) method (Maxwell equations) – Utzmann et al. [AIAA, 2006], L ′ eger et al. [AIAA, 2012], DG+FD (Acoustic) Hybrid FD + DG: Higher order hybrid WENO-FD+RKDG
BeiHang Outline University Introduction Introduction RKDG and WENO FD methods RKDG and WENO-FD methods Hybrid RKDG+WENO FD method Hybrid RKDG+WENO-FD method Numerical results Numerical results Conclusions and future work Conclusions and future work
RKDG methods Two dimensional hyperbolic conservation laws: ( ) ( ) ( ) ( ) 0 in 0 in (0 (0, ) ) u u f u f u g u g u T T t x y ( , ,0) ( , ) u x y u x y 0 Spatial discretization: The solution and test function space: p K k { ( , ) : ( , ) | ( )} V v x y v x y P h j j DG adopts a series of local basis over target cell: DG adopts a series of local basis over target cell: ( ) l { ( , ), 0,1,..., ; ( 1)( 2) / 2 1} v x y l K K k k The numerical solution can be written as: ( ) h l ( , , ) ( , , ) ( ) ( ) ( , ) ( , ) u u x y t x y t u t v u t v x y x y l l l
RKDG methods Multiply test functions and integrate over target cell: d d ( ) ( ) h l h h T l ( , ) ( ( ), ( )) ( , ) u v x y dxdy f u g u nv x y ds dt j j ( ) ( ) ( ) ( ) h l h l ( ( ( ( ) ) ( ( , ) ) ( ( ) ) ( ( , )) )) 0 0 f u f u v v x y x y g u g u v v x y dxdy x y dxdy x y j 0,..., l k where h ( ( , ) ) n n n x y On cell boundaries, the numerical solution is discontinuous, a numerical flux based on Riemann solution is used to replace the original flux: based on Riemann solution is used to replace the original flux: h h T ( ( ), ( )) ( , ) f u g u n h u u , n j Time discretization: third-order Runge-Kutta method
WENO methods WENO-FD WENO-FV (finite difference based WENO) (finite difference based WENO) (finite volume based WENO) (finite volume based WENO) Efficient for structured mesh Easy in treatment of complex boundaries boundaries Not applicable for unstructured mesh Costly and troublesome for maintaining higher order for Difficult in treatment of complex unstructured mesh boundaries WENO-FV has computational cost 4 times (2D)/9 times (3D)larger than WENO-FD for 3 rd order accuracy! ( ) g y
WENO-FD schemes Two dimensional hyperbolic conservation laws: ( ) ( ) ( ) ( ) 0 in 0 in (0, ) (0 ) u u f u f u g u g u T T t x y ( , ,0) ( , ) u x y u x y 0 Spatial discretization: For a WENO-FD scheme, uniform grid is required and solve directly using a conservative approximation to the space derivative: a conservative approximation to the space derivative: du 1 1 ˆ ˆ , i j ˆ ˆ ( ( ) ) ( ( ) ) 0 0 f f f f g g g g 1 1 1 1 dt x y , , , , i j i j i j i j 2 2 2 2 ˆ ˆ , , f f g g Th The numerical fluxes are obtained by one dimensional WENO-FD i l fl bt i d b di i l WENO FD 1 1 1 1 , , i j i j 2 2 approximation procedure.
WENO-FD schemes One dimensional WENO-FD procedure : (5 th -order WENO-FD) WENO construct polynomial q (x) on each candidate stencil S 0 ,S 1 ,S 2 and use the convex combination of all candidate stencils to achieve high order accurate convex combination of all candidate stencils to achieve high order accurate. 0 0 0 0 3 ( ) q a f a f a f O x 1 1 2 2 1 3 j j j j 2 1 1 1 1 3 ( ( ) ) q q a f a f a f a f a f a f O O x x 1 1 j 2 2 j 1 3 j j 2 2 2 2 2 3 ( ) q a f a f a f O x 1 1 2 2 1 3 j j j j 2 The numerical flux for 5 th order WENO-FD: ˆ 0 1 2 f d q d q d q 1 0 1 1 1 2 1 j j j j 2 2 2 2 2 2 2 2
WENO-FD schemes One dimensional WENO-FD procedure: (5 th -order WENO-FD) Classical WENO schemes use the smooth indicator(Jiang and Shu JCP,1996) Classical WENO schemes use the smooth indicator(Jiang and Shu JCP1996) of each stencil as follows: 1 l k r ( ) q x x j 1/2 2 l 1 2 ( ) x dx k k l l x x 1/2 j l 1 The nonlinear weights are given by: d d k k 0,1,..., 1 w k r k 1 k 2 r ( ) k s 0 s The numerical flux for 5 th order WENO-FD: ˆ 0 1 2 f f w q w q w q w q w q w q 1 0 1 1 1 2 1 j j j j 2 2 2 2
Summary RKDG RKDG WENO ‐ FD WENO ‐ FD Well in handling complex g p Highly efficient in structured grid g y g Advantage Advantage geometries Expensive in computational Only in uniform mesh and hard in costs and storage Weakness handling complex geometry requirements requirements
BeiHang Outline University Introduction Introduction RKDG and WENO FD methods RKDG and WENO-FD methods Hybrid RKDG+WENO FD method Hybrid RKDG+WENO-FD method Numerical results Numerical results Conclusions and future work Conclusions and future work
Multidomain hybrid RKDG+WENO-FD method RKDG+WENO-FD method Couple RKDG and WENO FD based on domain decomposition 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 Cut ‐ cell approach Hybrid mesh approach
RKDG+WENO-FD method: structured meshes RKDG+WENO-FD method for one dimensional conservation laws dimensional conservation laws Conservative coupling method: Conservative coupling method: Non-conservative coupling method: Non conservative coupling method:
RKDG+WENO-FD method: Construction of interface flux Construction of WENO flux at the interface: 1 1. Deploy ghost nodes at the DG domain Deploy ghost nodes at the DG domain 2. Compute the point value at each ghost points via DG solution 3. Obtain the WENO flux at the cell interface J+1/2
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