扩散方程九点格式中节点未知量的一种新的插值算法
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摘要
本文研究了二维扩散方程九点格式中节点辅助未知量的插值问题.利用多点通量逼近的边未知量插值算法和一个特殊的极限技巧,获得了节点辅助未知量的一个新的插值算法,并在给定假设下严格分析了该算法中局部线性系统的可解性.新算法满足线性精确准则,具有较高的精度.
In this paper, we discuss the interpolation problem for nodal auxiliary unknowns in nine point scheme for 2 D diffusion problems. By applying a special limit technique to the edge interpolation algorithm in multipoint flux approximation, we obtain a new nodal interpolation algorithm. Moreover, the solvability of the local system in the interpolation algorithm is analyzed rigorously under certain assumptions. The new algorithm satisfies linearity preserving criterion and has a second-order accuracy.
In this paper, we discuss the interpolation problem for nodal auxiliary unknowns in nine point scheme for 2 D diffusion problems. By applying a special limit technique to the edge interpolation algorithm in multipoint flux approximation, we obtain a new nodal interpolation algorithm. Moreover, the solvability of the local system in the interpolation algorithm is analyzed rigorously under certain assumptions. The new algorithm satisfies linearity preserving criterion and has a second-order accuracy.
引文
[1]李德元,水鸿寿,汤敏君.关于非矩形网格上的二维抛物方程的差分格式问.数值计算与计算机应用,1980, 1(4):217-224.
[2] Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids[J]. Comput.Geosci., 2002, 6(3):405-432.
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[4] Li Ronghua, Chen Zhongying, Wu Wei. The generalized difference methods for partial differential equations[M]. New York:Marcel Dekker, 2000.
[5] Herbin R, Hubert F. Benchmark on discretization schemes for anisotropic diffusion problems on general grids[A]. Eymard R, Herard J M. Finite volumes for complex applications V[C]. Wiley,2008:659-692.
[6] Eymard R, Henry G, Herbin R, et al. 3D Benchmark on discretization schemes for anisotropic diffusion problems on general grids[A]. Jaroslav Fort, Jiri F(u|¨)rst, Jan Halama, Raphaele Herbin,Florence Hubert. Finite volumes for complex applications VI problems&perspectives[C]. Berlin,Heidelberg:Springer, 2011:895-930.
[7] Droniou J. Finite volume schemes for diffusion equations:introduction to and review of modern methods[J]. Math. Model. Meth. Appl. Sci., 2014, 24(8):1575-1619.
[8] Camier J S, Hermeline F. A monotone nonlinear finite volume method for approximating diffusion operators on general meshes[J]. Intern. J. Numer. Meth. Engin., 2016, 107(6):496-519.
[9] Zhang Xiaoping, Su Shuai, Wu Jiming. A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids[J]. J. Comput. Phys., 2017, 344:419-436.
[10] Pei Wenbing. The construction of simulation algorithms for laser fusion[J]. Commun. Comput. Phys.,2007, 2(2):255-270.
[11] Zhang Yang, Wu Jiming, Dai Zihuan, et al. Computational investigation of the magneto-RayleighTaylor instability in Z-pinch implosions[J]. Phys. Plasmas, 2010, 17(4):042702.
[12] Lipnikov K, Shashkov M, Svyatskiy D, Vassilevski Y. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J]. J. Comput. Phys.,2007, 227(1):492-512.
[13] Coudiere Y, Vila J P, Villedieu P. Convergence rate of a finite volume scheme for a two-dimensional diffusion convection problem[J]. Math. Model. Numer. Anal., 1999, 33(3):493-516.
[14] Gao Zhiming, Wu Jiming. A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes[J]. Int. J. Numer. Meth. Fluids, 2011, 67(12):2157-2183.
[15] Wu Jiming, Dai Zihuan, Gao Zhiming, Yuan Guangwei. Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes[J]. J. Comput. Phys., 2010, 229(9):3382-3401.
[2] Aavatsmark I. An introduction to multipoint flux approximations for quadrilateral grids[J]. Comput.Geosci., 2002, 6(3):405-432.
[3] Shashkov M, Steinberg S. Solving diffusion equations with rough coefficients in rough grids[J]. J.Comput. Phys., 1996, 129(2):383-405.
[4] Li Ronghua, Chen Zhongying, Wu Wei. The generalized difference methods for partial differential equations[M]. New York:Marcel Dekker, 2000.
[5] Herbin R, Hubert F. Benchmark on discretization schemes for anisotropic diffusion problems on general grids[A]. Eymard R, Herard J M. Finite volumes for complex applications V[C]. Wiley,2008:659-692.
[6] Eymard R, Henry G, Herbin R, et al. 3D Benchmark on discretization schemes for anisotropic diffusion problems on general grids[A]. Jaroslav Fort, Jiri F(u|¨)rst, Jan Halama, Raphaele Herbin,Florence Hubert. Finite volumes for complex applications VI problems&perspectives[C]. Berlin,Heidelberg:Springer, 2011:895-930.
[7] Droniou J. Finite volume schemes for diffusion equations:introduction to and review of modern methods[J]. Math. Model. Meth. Appl. Sci., 2014, 24(8):1575-1619.
[8] Camier J S, Hermeline F. A monotone nonlinear finite volume method for approximating diffusion operators on general meshes[J]. Intern. J. Numer. Meth. Engin., 2016, 107(6):496-519.
[9] Zhang Xiaoping, Su Shuai, Wu Jiming. A vertex-centered and positivity-preserving scheme for anisotropic diffusion problems on arbitrary polygonal grids[J]. J. Comput. Phys., 2017, 344:419-436.
[10] Pei Wenbing. The construction of simulation algorithms for laser fusion[J]. Commun. Comput. Phys.,2007, 2(2):255-270.
[11] Zhang Yang, Wu Jiming, Dai Zihuan, et al. Computational investigation of the magneto-RayleighTaylor instability in Z-pinch implosions[J]. Phys. Plasmas, 2010, 17(4):042702.
[12] Lipnikov K, Shashkov M, Svyatskiy D, Vassilevski Y. Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J]. J. Comput. Phys.,2007, 227(1):492-512.
[13] Coudiere Y, Vila J P, Villedieu P. Convergence rate of a finite volume scheme for a two-dimensional diffusion convection problem[J]. Math. Model. Numer. Anal., 1999, 33(3):493-516.
[14] Gao Zhiming, Wu Jiming. A linearity-preserving cell-centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes[J]. Int. J. Numer. Meth. Fluids, 2011, 67(12):2157-2183.
[15] Wu Jiming, Dai Zihuan, Gao Zhiming, Yuan Guangwei. Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes[J]. J. Comput. Phys., 2010, 229(9):3382-3401.