Lax-Wendroff时间离散的自适应间断有限元方法求解三维可压缩欧拉方程
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摘要
应用自适应LWDG方法求解三维双曲守恒律方程组,与传统的二阶RKDG方法相比,该方法具有计算量小和精度高的特点.给出一种自适应策略,其中均衡折中策略适用于非相容四面体网格.将二维情形下的后验误差指示子推广到三维双曲守恒律方程组中,数值实验证明了方法的有效性.
We present a Lax-Wendroff discontinuous Galerkin( LWDG) method combining with adaptive mesh refinement( AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method( RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.
We present a Lax-Wendroff discontinuous Galerkin( LWDG) method combining with adaptive mesh refinement( AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method( RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.
引文
[1]Reed W H,Hill T R.Triangular mesh methods for the neutron transport equation[R].Los Alamos Scientific Laboratory Report LA-UR-73-479,1973.
[2]Cockburn B,Shu C W.TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II:General framework[J].Math Comp,1989,52:411-435.
[3]Bassi F,Rebay S.A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations[J].J Comput Phys,1997,131:267-279.
[4]Flaherty J E,Loy R M,Shephard M S,Szymanski B K,Teresco J D,Ziantz L H.Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws[J].Journal of Parallel and Distributed Computing,1997,47:139-152.
[5]张德良.计算流体力学教程[M].北京:高等教育出版社,2010.
[6]Babuska I,Zlamal M.Noncomforming elements in the finite element method with penalty[J].SIAM J Numer Anal,1973,10(5):863-875.
[7]Yang X F,James J,Lowengrub J,Zheng X M,Cristini V.An adaptive coupled level-set/volumeof-fluid interface capturing method for unstructured triangular grids[J].J Comput Phys,2006,217:364-394.
[8]Xu Y,Yu X J.Adaptive discontinuous Galerkin methods for hyperbolic conservation laws[J].Chinese J Comput Phys,2009,26(2):159-168.
[9]Wu D,Yu X J.Adaptive discontinuous Galerkin methods for three-dimensional Euler equations[J].Computer Physics Communications,2011,182(9):1771-1775.
[10]Qiu J,Dumbser M,Shu C W.The discontinuous Galerkin method with Lax-Wendroff type time discretizations[J].Comput Methods Appl Mech Engrg,2005,194:4528-4543.
[11]Tan S R,Shu C W.Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws[J].J Comput Phys,2010,229:8144-8166.
[12]吴迪,蔚喜军,徐云.局部时间长间断有限元方法求解三维欧拉方程[J].计算物理,2011,28(1):1-9.
[13]吴迪,蔚喜军.自适应间断有限元方法求解三维欧拉方程[J].计算物理,2010,27(4):492-500.
[14]S·li E,Houston P.Adaptive finite element approximation of hyperbolic problems[C]∥Barth T,Deconinck H,eds.Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics.LNCSE,Springer Verlag,2002,25:269-344.
[15]Adjerid S,Devine K D,Flaherty J E,Krivodouova L.A posteriori error estimation for discontinuous Galerkin solutions of hyperholic problems[J].Comput Methods Appl Mech Engrg,2002,191:1097-1112.
[2]Cockburn B,Shu C W.TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II:General framework[J].Math Comp,1989,52:411-435.
[3]Bassi F,Rebay S.A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations[J].J Comput Phys,1997,131:267-279.
[4]Flaherty J E,Loy R M,Shephard M S,Szymanski B K,Teresco J D,Ziantz L H.Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws[J].Journal of Parallel and Distributed Computing,1997,47:139-152.
[5]张德良.计算流体力学教程[M].北京:高等教育出版社,2010.
[6]Babuska I,Zlamal M.Noncomforming elements in the finite element method with penalty[J].SIAM J Numer Anal,1973,10(5):863-875.
[7]Yang X F,James J,Lowengrub J,Zheng X M,Cristini V.An adaptive coupled level-set/volumeof-fluid interface capturing method for unstructured triangular grids[J].J Comput Phys,2006,217:364-394.
[8]Xu Y,Yu X J.Adaptive discontinuous Galerkin methods for hyperbolic conservation laws[J].Chinese J Comput Phys,2009,26(2):159-168.
[9]Wu D,Yu X J.Adaptive discontinuous Galerkin methods for three-dimensional Euler equations[J].Computer Physics Communications,2011,182(9):1771-1775.
[10]Qiu J,Dumbser M,Shu C W.The discontinuous Galerkin method with Lax-Wendroff type time discretizations[J].Comput Methods Appl Mech Engrg,2005,194:4528-4543.
[11]Tan S R,Shu C W.Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws[J].J Comput Phys,2010,229:8144-8166.
[12]吴迪,蔚喜军,徐云.局部时间长间断有限元方法求解三维欧拉方程[J].计算物理,2011,28(1):1-9.
[13]吴迪,蔚喜军.自适应间断有限元方法求解三维欧拉方程[J].计算物理,2010,27(4):492-500.
[14]S·li E,Houston P.Adaptive finite element approximation of hyperbolic problems[C]∥Barth T,Deconinck H,eds.Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics.LNCSE,Springer Verlag,2002,25:269-344.
[15]Adjerid S,Devine K D,Flaherty J E,Krivodouova L.A posteriori error estimation for discontinuous Galerkin solutions of hyperholic problems[J].Comput Methods Appl Mech Engrg,2002,191:1097-1112.