一种新的混合有限体积元法求解一维多孔介质问题
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摘要
针对一维多孔介质问题用标准混合有限体积元法求解时会出现数值解波阵面不能向前传播的现象,提出一种新的混合有限体积元法求解退化问题,其中流变量仅包含原始变量对空间变量的导数.结果表明,该方法可避免数值解波阵面不能向前传播的现象,并能很好地捕捉数值解界面.数值实验验证了该方法的有效性.
For the one-dimensional porous medium problem,wave front of the numerical solution could not propagate forward when the standard mixed finite volume element method was used to solve them,we proposed a new mixed finite volume element method for solving the degradation problem,in which the flux variable only included the derivative of the original variable to spacial variable.The results show that the method can avoid the phenomenon that wave front of the numerical solution can not propagate forward,and can capture the interface of numerical solution well.The validity of the method is verified by numerical experiments.
For the one-dimensional porous medium problem,wave front of the numerical solution could not propagate forward when the standard mixed finite volume element method was used to solve them,we proposed a new mixed finite volume element method for solving the degradation problem,in which the flux variable only included the derivative of the original variable to spacial variable.The results show that the method can avoid the phenomenon that wave front of the numerical solution can not propagate forward,and can capture the interface of numerical solution well.The validity of the method is verified by numerical experiments.
引文
[1]NOCHETTO R H,VERDI C.Approximation of Degenerate Parabolic Problems Using Numerical Integration[J].SIAM Journal on Numerical Analysis,1988,25(4):784-814.
[2]EBMEYER C.Error Estimates for a Class of Degenerate Parabolic Equations[J].SIAM Journal on Numerical Analysis,1998,35(3):1095-1112.
[3]LISKA R,SHASHKOV M.Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions for Elliptic Problems[J].Communications in Computational Physics,2008,3(4):852-877.
[4]ZHANG Qiang,WU Zilong.Numerical Simulation for Porous Medium Equation by Local Discontinuous Galerkin Finite Element Method[J].Journal of Scientific Computing,2009,38(2):127-148.
[5]NGO C,HUANG Weizhang.A Study on Moving Mesh Finite Element Solution of the Porous Medium Equation[J].Journal of Computational Physics,2017,331:357-380.
[6]LIPNIKOV K,MANZINI G,MOULTON J D,et al.The Mimetic Finite Difference Method for Elliptic and Parabolic Problems with a Staggered Discretization of Diffusion Coefficient[J].Journal of Computational Physics,2016,305:111-126.
[7]陈国芳,吴丹,吕俊良.一种非标准的混合有限元法求解一维退化非线性抛物问题[J].吉林大学学报(理学版),2017,55(6):1352-1358.(CHEN Guofang,WU Dan,LJunliang.A Nonstandard Mixed Finite Element Method for Solving One-Dimensional Degenerate Nonlinear Parabolic Problem[J].Journal of Jilin University(Science Edition),2017,55(6):1352-1358.)
[8]YUE Jingyan,YUAN Guangwei.Picard-Newton Iterative Method with Time Step Control for Multimaerial Non-equilibrium Radiation Diffusion Problem[J].Communications in Computational Physics,2011,10(4):844-866.
[9]伍卓群,赵俊宁,尹景学,等.非线性扩散方程[M].长春:吉林大学出版社,1996.(WU Zhuoqun,ZHAO Junning,YIN Jingxue,et al.Nonlinear Diffusion Equations[M].Changchun:Jilin University Press,1996.)
[10]尹景学.非线性扩散方程广义解的有限传播速度性质[J].数学学报,1991,34(3):360-364.(YIN Jingxue.The Property of Finite Speed of Propagation of Generalized Solutions to Nonlinear Diffusion Equations[J].Acta Mathematica Sinica,1991,34(3):360-364.)
[11]WU Dan,YUE Jingyan,YUAN Guangwei,et al.Finite Volume Element Approximation for Nonlinear Diffusion Problems with Degenerate Diffusion Coefficients[J].Applied Numerical Mathematics,2019,140:23-47.
[12]LV Junliang,YUAN Guangwei,YUE Jingyan.Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems[J].Numerical Mathematics:Theory,Methods and Applications,2018,11(3):413-436.
[2]EBMEYER C.Error Estimates for a Class of Degenerate Parabolic Equations[J].SIAM Journal on Numerical Analysis,1998,35(3):1095-1112.
[3]LISKA R,SHASHKOV M.Enforcing the Discrete Maximum Principle for Linear Finite Element Solutions for Elliptic Problems[J].Communications in Computational Physics,2008,3(4):852-877.
[4]ZHANG Qiang,WU Zilong.Numerical Simulation for Porous Medium Equation by Local Discontinuous Galerkin Finite Element Method[J].Journal of Scientific Computing,2009,38(2):127-148.
[5]NGO C,HUANG Weizhang.A Study on Moving Mesh Finite Element Solution of the Porous Medium Equation[J].Journal of Computational Physics,2017,331:357-380.
[6]LIPNIKOV K,MANZINI G,MOULTON J D,et al.The Mimetic Finite Difference Method for Elliptic and Parabolic Problems with a Staggered Discretization of Diffusion Coefficient[J].Journal of Computational Physics,2016,305:111-126.
[7]陈国芳,吴丹,吕俊良.一种非标准的混合有限元法求解一维退化非线性抛物问题[J].吉林大学学报(理学版),2017,55(6):1352-1358.(CHEN Guofang,WU Dan,LJunliang.A Nonstandard Mixed Finite Element Method for Solving One-Dimensional Degenerate Nonlinear Parabolic Problem[J].Journal of Jilin University(Science Edition),2017,55(6):1352-1358.)
[8]YUE Jingyan,YUAN Guangwei.Picard-Newton Iterative Method with Time Step Control for Multimaerial Non-equilibrium Radiation Diffusion Problem[J].Communications in Computational Physics,2011,10(4):844-866.
[9]伍卓群,赵俊宁,尹景学,等.非线性扩散方程[M].长春:吉林大学出版社,1996.(WU Zhuoqun,ZHAO Junning,YIN Jingxue,et al.Nonlinear Diffusion Equations[M].Changchun:Jilin University Press,1996.)
[10]尹景学.非线性扩散方程广义解的有限传播速度性质[J].数学学报,1991,34(3):360-364.(YIN Jingxue.The Property of Finite Speed of Propagation of Generalized Solutions to Nonlinear Diffusion Equations[J].Acta Mathematica Sinica,1991,34(3):360-364.)
[11]WU Dan,YUE Jingyan,YUAN Guangwei,et al.Finite Volume Element Approximation for Nonlinear Diffusion Problems with Degenerate Diffusion Coefficients[J].Applied Numerical Mathematics,2019,140:23-47.
[12]LV Junliang,YUAN Guangwei,YUE Jingyan.Nonnegativity-Preserving Repair Techniques for the Finite Element Solutions of Degenerate Nonlinear Parabolic Problems[J].Numerical Mathematics:Theory,Methods and Applications,2018,11(3):413-436.