各向异性扩散问题Kershaw格式的守恒保正修复算法
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摘要
针对各向异性扩散方程Kershaw格式的数值解在正交网格及扭曲网格上计算出负的现象,给出一种守恒的保正修复算法(CENZ),该算法对简单遇负置零(ENZ)方法进行改进,使修复后的数值解不仅具有非负性,而且保持法向通量的局部守恒性.数值算例表明,该方法不受计算网格类型和扩散系数各向异性比的限制,可用于对任意违背单调性(或保正性)的有限体积格式数值解的修复.
Kershaw scheme is not positivity-preserving. Negative values emerge in numerical simulation for anisotropic diffusion equations on both orthogonal and distorted meshes. A conservative enforcing negative value to zero( CENZ) algorithm is proposed,which is an improvement of traditional method. It not only repairs numerical solution nonnegative,but also keeps local conservation of energy fluxes. Numerical examples demonstrate that the method is not limited by anisotropic ratio of diffusion coefficients. The algorithm can be used for numerical solution of finite volume schemes which violate monotony or positivity-preserving.
Kershaw scheme is not positivity-preserving. Negative values emerge in numerical simulation for anisotropic diffusion equations on both orthogonal and distorted meshes. A conservative enforcing negative value to zero( CENZ) algorithm is proposed,which is an improvement of traditional method. It not only repairs numerical solution nonnegative,but also keeps local conservation of energy fluxes. Numerical examples demonstrate that the method is not limited by anisotropic ratio of diffusion coefficients. The algorithm can be used for numerical solution of finite volume schemes which violate monotony or positivity-preserving.
引文
[1]KERSHAW D S.Differencing of the diffusion equation in Lagrangian hydrodynamic codes[J].J Comput Phys,1981,39:375-395.
[2]李德元,水鸿寿,汤敏君.关于非矩形网格上的二维抛物型方程的差分格式[J].数值计算与计算机应用,1980,1:217-224.
[3]MOREL J E,DENDY J E,HALL M L,et al.A cell centered Lagrangian-mesh diffusion differencing scheme[J].J Comput Phys,1992,103:286-299.
[4]AAVATSMARK I.An introduction to multipoint flux approximation for quadrilateral grids[J].Comput Geosci,2002,6:405-432.
[5]AAVATSMARK I.Comparison of monotonicity for some multipoint flux approximation methods.Finite Volumes for Complex Applications[M].New York:Wiley,2008,5:19-34.
[6]AAVATSMARK I,EIGESTAD G,MALLISON B,et al.A compact multipoint flux approximation method with improved robustness[J].Numer Meth Part D E,2008,24:1329-1360.
[7]POTIER C L.Finite volume monotone scheme for highly anisotropic diffusion operations on unstructrued triangular meshes[J].CR Acad Sci I-Math,2005,341(12):787-792.
[8]YUAN G W,SHENG Z Q.Monotone finite volume schemes for diffusion equations on polygonal meshes[J].J Comput Phys,2008,227:6288-6312.
[9]LIPNIKOV K,SHASHKOV M,SVYATSKIY D,et al.Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J].J Comput Phys,2007,227:492-512.
[10]CANCES C,CATHALA M,POTIER C L.Monotone corrections for generic cell-centered finite volume approximations of anistropic diffusion equations[J].Nummer Math,2013,198:1638-1646.
[11]POTIER L C.A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators[J].Int JFinite Volumes,2009,6:1-20.
[12]ARICO C,TUCCIARELLI T.Monotonic solution of heterogeneous anisotropic diffusion problems[J].J Comput Phys,2013,252:219-249.
[13]KUZMIN D,SHASHKOV M J,SVYATSKIY D.A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems[J].J Comput Phys,2009,228:3448-3463.
[14]HOTEIT H,MOSE R,PHILIPPE B,et al.The maximum principle violations of the mixed-hybrid finite element method applied to diffusion equations[J].Numer Meth Eng,2002,55(12):1373-1390.
[15]SHENG Z Q,YUAN G W.The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes[J].J Comput Phys,2011,230(7):2588-2604.
[16]PAL M,EDWARDS G.Non-linear flux-splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients[J].Int J Numer Meth Fluids,2011,66:299-323.
[17]LISKA R,SHASHKOV M.Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems[J].Commun Comput Phys,2008,3(4):852-877.
[18]WANG S,YUAN G W,LI Y H,et al.Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems[J].Int J Numer Method Fluid,2012,70(9):1188-1205.
[19]KUZMIN D,SHASKOV M J,SVYATSKIY D.A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems[J].J Comput Phys,2009,228:3448-3463.
[20]YAO Y Z,YUAN G W.Enforcing positivity with conservation for nine-point scheme of nonlinear diffusion equations[J].Comput Methods Appl Mech Engrg,2012,223-224:161-172.
[21]曾清红,裴文兵,成娟,勇珩.多块结构网格上的Kershaw扩散格式[J].计算物理,2011,28:641-648.
[2]李德元,水鸿寿,汤敏君.关于非矩形网格上的二维抛物型方程的差分格式[J].数值计算与计算机应用,1980,1:217-224.
[3]MOREL J E,DENDY J E,HALL M L,et al.A cell centered Lagrangian-mesh diffusion differencing scheme[J].J Comput Phys,1992,103:286-299.
[4]AAVATSMARK I.An introduction to multipoint flux approximation for quadrilateral grids[J].Comput Geosci,2002,6:405-432.
[5]AAVATSMARK I.Comparison of monotonicity for some multipoint flux approximation methods.Finite Volumes for Complex Applications[M].New York:Wiley,2008,5:19-34.
[6]AAVATSMARK I,EIGESTAD G,MALLISON B,et al.A compact multipoint flux approximation method with improved robustness[J].Numer Meth Part D E,2008,24:1329-1360.
[7]POTIER C L.Finite volume monotone scheme for highly anisotropic diffusion operations on unstructrued triangular meshes[J].CR Acad Sci I-Math,2005,341(12):787-792.
[8]YUAN G W,SHENG Z Q.Monotone finite volume schemes for diffusion equations on polygonal meshes[J].J Comput Phys,2008,227:6288-6312.
[9]LIPNIKOV K,SHASHKOV M,SVYATSKIY D,et al.Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes[J].J Comput Phys,2007,227:492-512.
[10]CANCES C,CATHALA M,POTIER C L.Monotone corrections for generic cell-centered finite volume approximations of anistropic diffusion equations[J].Nummer Math,2013,198:1638-1646.
[11]POTIER L C.A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators[J].Int JFinite Volumes,2009,6:1-20.
[12]ARICO C,TUCCIARELLI T.Monotonic solution of heterogeneous anisotropic diffusion problems[J].J Comput Phys,2013,252:219-249.
[13]KUZMIN D,SHASHKOV M J,SVYATSKIY D.A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems[J].J Comput Phys,2009,228:3448-3463.
[14]HOTEIT H,MOSE R,PHILIPPE B,et al.The maximum principle violations of the mixed-hybrid finite element method applied to diffusion equations[J].Numer Meth Eng,2002,55(12):1373-1390.
[15]SHENG Z Q,YUAN G W.The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes[J].J Comput Phys,2011,230(7):2588-2604.
[16]PAL M,EDWARDS G.Non-linear flux-splitting schemes with imposed discrete maximum principle for elliptic equations with highly anisotropic coefficients[J].Int J Numer Meth Fluids,2011,66:299-323.
[17]LISKA R,SHASHKOV M.Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems[J].Commun Comput Phys,2008,3(4):852-877.
[18]WANG S,YUAN G W,LI Y H,et al.Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems[J].Int J Numer Method Fluid,2012,70(9):1188-1205.
[19]KUZMIN D,SHASKOV M J,SVYATSKIY D.A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems[J].J Comput Phys,2009,228:3448-3463.
[20]YAO Y Z,YUAN G W.Enforcing positivity with conservation for nine-point scheme of nonlinear diffusion equations[J].Comput Methods Appl Mech Engrg,2012,223-224:161-172.
[21]曾清红,裴文兵,成娟,勇珩.多块结构网格上的Kershaw扩散格式[J].计算物理,2011,28:641-648.