发展型方程的混合有限体积元方法及数值模拟
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摘要
混合有限体积元方法是将混合有限元方法和有限体积元方法相结合的一种数值方法,该方法又称为混合控制体积方法,最早是由Russell于1995年在求解一类二阶线性椭圆型问题时提出,随后Cai和Jones等人通过数值算例验证该方法的有效性,Chou和Kwak等人在该方法的理论分析方面做了大量的工作.该方法具有如下优点：采用混合元思想引入辅助变量(如梯度函数,流函数等)将高阶问题转化为低阶问题,降低对有限元空间的光滑度的要求；易于处理复杂区域和边界条件,具有有限体积元方法的格式简单性；方法的定义采用混合变分形式,可以从混合变分形式出发进行理论分析；计算量比混合有限元方法小,收敛阶和混合有限元方法相同；能够保持某些物理量(如质量,动量)的局部守恒性质.
Rui和Lu于2005年将扩展混合元方法和有限体积元方法相结合,对一类二阶椭圆问题构造了矩形网格剖分下的扩展混合有限体积元方法,该方法继承了扩展元方法和有限体积元方法的优势,可以同时数值计算三个未知变量.目前关于此类方法的研究主要是基于矩形网格剖分,而在三角网格剖分下的研究还非常少,本文主要是在三角网格剖分下将此类方法求解含对流项Sobolev方程,并给出误差分析和数值模拟.
近年来随着解决复杂实际数学物理问题的需要,对数值方法在计算效率上也有了更高的要求.本文结合分裂思想对扩展混合有限体积元方法进行简化,提出了一类新型的分裂扩展混合有限体积元方法.该方法在数值计算时可以先求解两个方程的耦合系统得到两个变量的数值解,再利用这两个数值解求解第三个方程得到第三个变量的数值解,这种办法在很大程度上降低了线性方程组的规模从而大幅度的减少计算时间.
本文应用混合有限体积元、扩展混合有限体积元以及分裂扩展混合有限体积元方法从理论分析和数值计算两个方面对几类发展型方程进行研究,由于每一类方程都有不同的特点,从而所构造的数值格式也不相同,而且需要根据每一类方程的特点进行相应的理论分析和数值实验.在第一章中简单介绍一下混合有限元方法和混合有限体积元方法的特点和发展现状.
在第二章到第四章中应用混合有限体积元方法研究三类发展型方程.其中在第二章研究了一维正则长波方程的混合有限体积元方法.通过引入一维网格剖分下的迁移算子,构造了半离散、非线性和线性向后Euler全离散混合有限体积元格式,利用椭圆投影和L2正交投影算子给出三种离散格式的最优阶误差估计,最后通过数值算例验证格式的有效性和收敛精度.第三章和第四章应用混合有限体积元方法在三角网格剖分下数值求解一类二维伪双曲型方程和非线性阻尼Sine-Gordon方程.选用最低阶Raviart-Thomas有限元空间和分片常函数空间作为解函数空间,并引入迁移算子γh将最低阶Raviart-Thomas有限元空间映射成试探函数空间,对两类方程分别构造了半离散和关于时间隐式的全离散混合有限体积元格式.通过引入广义混合有限体积投影得到半离散和全离散格式的最优误差估计,最后对两类方程分别给出一些数值结果验证了该方法的可行性.
第五章将扩展混合元和混合有限体积元方法相结合构造了一类含对流项Sobolev方程的初边值问题的扩展混合有限体积元方法.该方法引入辅助变量λ=-▽u和σ=-(a▽u+6Vut),将原问题降为一阶微分方程系统,选用最低阶Raviart-Thomas有限元空间作为变量λ和σ的解函数空间,并使用分片常函数空间作为u的解函数空间,利用迁移算子γh在三角网格剖分下构造了半离散和向后Euler全离散的扩展混合有限体积元格式.应用微分方程理论证明了半离散格式解的存在唯一性,利用迁移算子的性质和扩展混合有限体积投影得到半离散和全离散格式的最优阶误差估计.最后给出数值算例验证了方法的可行性和理论分析的正确性.
第六章将扩展混合有限体积元方法和分裂思想相结合,引入和第五章一样的辅助变量λ和σ,构造了含对流项Sobolev方程的一种新型分裂扩展混合有限体积元格式.这一格式与扩展混合有限体积元格式的区别在于：扩展混合有限体积元格式需要同时求解三个方程的的耦合系统,从而在数值计算过程中生成的线性方程组的系数矩阵规模比较大,而此格式在数值计算时可以先求解两个方程的耦合系统得到变量λ和σ的数值解,再利用这两个数值解求解第三个方程得到变量u的数值解,这种办法在很大程度上降低了线性方程组的规模从而大幅度的减少计算时间.最后给出一些数值结果来验证该方法的有效性和理论结果的正确性.
第七章研究了一类抛物型积分微分方程的初边值问题的分裂扩展混合有限体积元方法.引入辅助变量λ(x,t)=-(?)u(x,t)和σ(x,t)=-(a(x)(?)u(x,t)+∫0t(x,t,τ)(?)u(x,τ)dτ),利用迁移算子γh构造了半离散和向后Euler全离散分裂扩展混合有限体积元格式,其中在全离散格式中利用左矩形数值积分公式离散积分项,利用向后Euler格式离散时间导数项.引入Volterra型扩展混合有限体积投影并利用迁移算子的性质得到了两种格式的最优阶误差估计,最后通过数值算例验证了理论分析结果.
Mixed finite volume element methods which combine mixed finite element methods with finite volume element methods, also called as mixed covolume methods, were first introduced by Russell in1995for solving a class of second-order linear elliptic type prob-lem. Soon afterwards, Cai and Jones et al illustrated the effectiveness of the proposed methods, Chou and Kwak et al had done a lot of work in the aspect of theoretical analysis. The methods have the following advantages:the methods use the idea of mixed methods and introduce auxiliary variables (such as gradient function, flux function etc.) to refor-mulate higher order problems as lower order problems, and then reduce the regularity requirement of the finite element spaces; it is easy to handle the complex regions and boundary conditions, and the methods have the simple format similar to finite volume element methods; the definition of the methods uses mixed variational form which is help-ful to make theoretical analysis; the computational effort is less than mixed finite element methods, and convergence accuracy is the same as the mixed finite element methods; the local conservation properties of some physical quantities (such as Mass, momentum etc.) are maintained. Based on the above features and advantages, mixed finite volume element methods have become the important numerical methods for solving differential equations.
Rui and Lu (in2005) combined expanded mixed methods with finite volume element methods, and constructed an expanded mixed finite volume element method based on the rectangular grids for a class of second-order elliptic problem. This method Inherits the advantages of expanded mixed methods and finite volume element methods, and can simultaneously calculate three unknown variables. The study of this method is mainly based on rectangular grids, however, the analysis on triangular grids is relatively few. In this thesis, we apply expanded mixed finite volume element method based on triangular girds to solving a class of Sobolev equation with convection term, and give the error analysis and numerical simulation.
In recent years, with the need to solve complex practical problems of mathematical physics, numerical methods have higher requirements on the computational efficiency. In this thesis, we simplify the expanded mixed finite volume element methods by combining with splitting idea, and propose a new splitting expanded mixed finite volume element. In the numerical simulation process, this method can first solve the coupling system of two equations to get the numerical solutions of the two variables, and then solve the third equation to get the numerical solution of the third variable, thus, greatly reduce the size of linear equations and the computational time.
In this thesis, we will apply the mixed finite volume element methods, expanded mixed finite volume element methods and splitting expanded mixed finite volume element methods to studying some evolution type equations from the aspects of the theoretical analysis and numerical calculation. Due to the different characteristics of each class of equations, the numerical formats we constructed are not the same, and the corresponding theoretical analysis and numerical experiments need to be carried out according to the characteristics of each class of equations. In Chapter1, the features and development status of the mixed finite element methods and mixed finite volume element methods are introduced briefly.
In Chapter2to Chapter4, we apply the mixed finite volume element method to solv-ing three classes of evolution type equations. In Chapter2, a mixed finite volume element method for one-dimensional regularized long wave equation is studied. The semidiscrete, nonlinear and linear backward Euler fully discrete schemes for the proposed problem are constructed by introducing the transfer operator based on one-dimensional grids. The optimal error estimates for three schemes are derived by using elliptic projection and L2-orthogonal projection operators. Finally, a numerical example is provided to verify the ef-fectiveness and convergence rate. In Chapter3and Chapter4, we apply the method based on triangular grids to numerically solving a class of two-dimensional pseudo-hyperbolic type equation and nonlinear damped Sine-Gordon equation. By using the lowest order Raviart-Thomas space and piecewise constant function space as the solution function spaces, and introducing the transfer operator γh which maps the lowest order Raviart-Thomas space into the test function space, both the semidiscrete and time-in-implicit fully discrete mixed finite volume element schemes are formulated for two proposed equations, respectively. Optimal error estimates are derived by introducing generalized mixed finite volume projection. Finally, some numerical results for two given equations are provided to test the effectiveness of the proposed schemes.
In Chapter5, an expanded mixed finite volume element procedure for a class of Sobolev equation with convection term is formulated by combining the expanded mixed methods with finite volume element methods. This method introduces two auxiliary vari-ables λ=-(>)u and σ=-(a(?)u+b(>)ut) to rewrite the original equation as the system of first-order differential equations, uses the lowest order Raviart-Thomas space as the solution function spaces of the variables λ and σ, and uses the piecewise constant func-tion space as the solution function space of the variable u. Then, both the semidiscrete and backward Euler fully discrete expanded mixed finite volume element schemes are con-structed based on triangular grids by using the transfer operator γh. The existence and uniqueness of semidiscrete scheme solutions are proved by applying the theory of differ-ential equations, and the optimal error estimates for the semidiscrete and fully discrete schemes are derived by using the properties of transfer operator and expanded mixed finite volume projection. Finally, a numerical example is given to illustrate the feasibility and the correctness of the theoretical results.
In Chapter6, we introduce two auxiliary variables λ and σ defined in Chapter5, and construct a new splitting expanded mixed finite volume element scheme for the Sobolev equation with convection term. The difference between this scheme and the expanded mixed finite volume element scheme is in that:expanded mixed finite volume element scheme needs to solve the coupling system of three equations, and then the size of the coefficient matrix in the process of numerical calculation is relatively large; however, the expanded splitting mixed finite volume element scheme can first solve the coupling system of two equations to get the numerical solutions of the variables λ and σ, and then solve the third equation to get the numerical solution of the variable u, thus, greatly reduce the size of linear equations and the computational time. Finally, some numerical results are given to illustrate the feasibility and the correctness of the theoretical results.
In Chapter7, a splitting expanded mixed finite volume element method for a class of parabolic type integro-differential equation is formulated. By using transfer operator γh and introducing auxiliary variables λ(x,t)=-(?)u(x,t) and σ(x, t)=-(a(x)(?)u(x,t)+∫0tk(x, t,τ)(?)u(x, τ)dτ), both the semidiscrete and backward Euler fully discrete splitting expanded mixed finite volume element schemes are constructed. In the fully discrete scheme, left rectangle quadrature rule and backward Euler scheme are applied to getting the discretization of the integral term and the time derivative term, respectively. The optimal error estimates for two schemes are derived by introducing the Volterra type expanded mixed finite volume projection and using the properties of the transfer operator. Finally, a numerical example is given to confirm the theoretical results.
Rui和Lu于2005年将扩展混合元方法和有限体积元方法相结合,对一类二阶椭圆问题构造了矩形网格剖分下的扩展混合有限体积元方法,该方法继承了扩展元方法和有限体积元方法的优势,可以同时数值计算三个未知变量.目前关于此类方法的研究主要是基于矩形网格剖分,而在三角网格剖分下的研究还非常少,本文主要是在三角网格剖分下将此类方法求解含对流项Sobolev方程,并给出误差分析和数值模拟.
近年来随着解决复杂实际数学物理问题的需要,对数值方法在计算效率上也有了更高的要求.本文结合分裂思想对扩展混合有限体积元方法进行简化,提出了一类新型的分裂扩展混合有限体积元方法.该方法在数值计算时可以先求解两个方程的耦合系统得到两个变量的数值解,再利用这两个数值解求解第三个方程得到第三个变量的数值解,这种办法在很大程度上降低了线性方程组的规模从而大幅度的减少计算时间.
本文应用混合有限体积元、扩展混合有限体积元以及分裂扩展混合有限体积元方法从理论分析和数值计算两个方面对几类发展型方程进行研究,由于每一类方程都有不同的特点,从而所构造的数值格式也不相同,而且需要根据每一类方程的特点进行相应的理论分析和数值实验.在第一章中简单介绍一下混合有限元方法和混合有限体积元方法的特点和发展现状.
在第二章到第四章中应用混合有限体积元方法研究三类发展型方程.其中在第二章研究了一维正则长波方程的混合有限体积元方法.通过引入一维网格剖分下的迁移算子,构造了半离散、非线性和线性向后Euler全离散混合有限体积元格式,利用椭圆投影和L2正交投影算子给出三种离散格式的最优阶误差估计,最后通过数值算例验证格式的有效性和收敛精度.第三章和第四章应用混合有限体积元方法在三角网格剖分下数值求解一类二维伪双曲型方程和非线性阻尼Sine-Gordon方程.选用最低阶Raviart-Thomas有限元空间和分片常函数空间作为解函数空间,并引入迁移算子γh将最低阶Raviart-Thomas有限元空间映射成试探函数空间,对两类方程分别构造了半离散和关于时间隐式的全离散混合有限体积元格式.通过引入广义混合有限体积投影得到半离散和全离散格式的最优误差估计,最后对两类方程分别给出一些数值结果验证了该方法的可行性.
第五章将扩展混合元和混合有限体积元方法相结合构造了一类含对流项Sobolev方程的初边值问题的扩展混合有限体积元方法.该方法引入辅助变量λ=-▽u和σ=-(a▽u+6Vut),将原问题降为一阶微分方程系统,选用最低阶Raviart-Thomas有限元空间作为变量λ和σ的解函数空间,并使用分片常函数空间作为u的解函数空间,利用迁移算子γh在三角网格剖分下构造了半离散和向后Euler全离散的扩展混合有限体积元格式.应用微分方程理论证明了半离散格式解的存在唯一性,利用迁移算子的性质和扩展混合有限体积投影得到半离散和全离散格式的最优阶误差估计.最后给出数值算例验证了方法的可行性和理论分析的正确性.
第六章将扩展混合有限体积元方法和分裂思想相结合,引入和第五章一样的辅助变量λ和σ,构造了含对流项Sobolev方程的一种新型分裂扩展混合有限体积元格式.这一格式与扩展混合有限体积元格式的区别在于：扩展混合有限体积元格式需要同时求解三个方程的的耦合系统,从而在数值计算过程中生成的线性方程组的系数矩阵规模比较大,而此格式在数值计算时可以先求解两个方程的耦合系统得到变量λ和σ的数值解,再利用这两个数值解求解第三个方程得到变量u的数值解,这种办法在很大程度上降低了线性方程组的规模从而大幅度的减少计算时间.最后给出一些数值结果来验证该方法的有效性和理论结果的正确性.
第七章研究了一类抛物型积分微分方程的初边值问题的分裂扩展混合有限体积元方法.引入辅助变量λ(x,t)=-(?)u(x,t)和σ(x,t)=-(a(x)(?)u(x,t)+∫0t(x,t,τ)(?)u(x,τ)dτ),利用迁移算子γh构造了半离散和向后Euler全离散分裂扩展混合有限体积元格式,其中在全离散格式中利用左矩形数值积分公式离散积分项,利用向后Euler格式离散时间导数项.引入Volterra型扩展混合有限体积投影并利用迁移算子的性质得到了两种格式的最优阶误差估计,最后通过数值算例验证了理论分析结果.
Mixed finite volume element methods which combine mixed finite element methods with finite volume element methods, also called as mixed covolume methods, were first introduced by Russell in1995for solving a class of second-order linear elliptic type prob-lem. Soon afterwards, Cai and Jones et al illustrated the effectiveness of the proposed methods, Chou and Kwak et al had done a lot of work in the aspect of theoretical analysis. The methods have the following advantages:the methods use the idea of mixed methods and introduce auxiliary variables (such as gradient function, flux function etc.) to refor-mulate higher order problems as lower order problems, and then reduce the regularity requirement of the finite element spaces; it is easy to handle the complex regions and boundary conditions, and the methods have the simple format similar to finite volume element methods; the definition of the methods uses mixed variational form which is help-ful to make theoretical analysis; the computational effort is less than mixed finite element methods, and convergence accuracy is the same as the mixed finite element methods; the local conservation properties of some physical quantities (such as Mass, momentum etc.) are maintained. Based on the above features and advantages, mixed finite volume element methods have become the important numerical methods for solving differential equations.
Rui and Lu (in2005) combined expanded mixed methods with finite volume element methods, and constructed an expanded mixed finite volume element method based on the rectangular grids for a class of second-order elliptic problem. This method Inherits the advantages of expanded mixed methods and finite volume element methods, and can simultaneously calculate three unknown variables. The study of this method is mainly based on rectangular grids, however, the analysis on triangular grids is relatively few. In this thesis, we apply expanded mixed finite volume element method based on triangular girds to solving a class of Sobolev equation with convection term, and give the error analysis and numerical simulation.
In recent years, with the need to solve complex practical problems of mathematical physics, numerical methods have higher requirements on the computational efficiency. In this thesis, we simplify the expanded mixed finite volume element methods by combining with splitting idea, and propose a new splitting expanded mixed finite volume element. In the numerical simulation process, this method can first solve the coupling system of two equations to get the numerical solutions of the two variables, and then solve the third equation to get the numerical solution of the third variable, thus, greatly reduce the size of linear equations and the computational time.
In this thesis, we will apply the mixed finite volume element methods, expanded mixed finite volume element methods and splitting expanded mixed finite volume element methods to studying some evolution type equations from the aspects of the theoretical analysis and numerical calculation. Due to the different characteristics of each class of equations, the numerical formats we constructed are not the same, and the corresponding theoretical analysis and numerical experiments need to be carried out according to the characteristics of each class of equations. In Chapter1, the features and development status of the mixed finite element methods and mixed finite volume element methods are introduced briefly.
In Chapter2to Chapter4, we apply the mixed finite volume element method to solv-ing three classes of evolution type equations. In Chapter2, a mixed finite volume element method for one-dimensional regularized long wave equation is studied. The semidiscrete, nonlinear and linear backward Euler fully discrete schemes for the proposed problem are constructed by introducing the transfer operator based on one-dimensional grids. The optimal error estimates for three schemes are derived by using elliptic projection and L2-orthogonal projection operators. Finally, a numerical example is provided to verify the ef-fectiveness and convergence rate. In Chapter3and Chapter4, we apply the method based on triangular grids to numerically solving a class of two-dimensional pseudo-hyperbolic type equation and nonlinear damped Sine-Gordon equation. By using the lowest order Raviart-Thomas space and piecewise constant function space as the solution function spaces, and introducing the transfer operator γh which maps the lowest order Raviart-Thomas space into the test function space, both the semidiscrete and time-in-implicit fully discrete mixed finite volume element schemes are formulated for two proposed equations, respectively. Optimal error estimates are derived by introducing generalized mixed finite volume projection. Finally, some numerical results for two given equations are provided to test the effectiveness of the proposed schemes.
In Chapter5, an expanded mixed finite volume element procedure for a class of Sobolev equation with convection term is formulated by combining the expanded mixed methods with finite volume element methods. This method introduces two auxiliary vari-ables λ=-(>)u and σ=-(a(?)u+b(>)ut) to rewrite the original equation as the system of first-order differential equations, uses the lowest order Raviart-Thomas space as the solution function spaces of the variables λ and σ, and uses the piecewise constant func-tion space as the solution function space of the variable u. Then, both the semidiscrete and backward Euler fully discrete expanded mixed finite volume element schemes are con-structed based on triangular grids by using the transfer operator γh. The existence and uniqueness of semidiscrete scheme solutions are proved by applying the theory of differ-ential equations, and the optimal error estimates for the semidiscrete and fully discrete schemes are derived by using the properties of transfer operator and expanded mixed finite volume projection. Finally, a numerical example is given to illustrate the feasibility and the correctness of the theoretical results.
In Chapter6, we introduce two auxiliary variables λ and σ defined in Chapter5, and construct a new splitting expanded mixed finite volume element scheme for the Sobolev equation with convection term. The difference between this scheme and the expanded mixed finite volume element scheme is in that:expanded mixed finite volume element scheme needs to solve the coupling system of three equations, and then the size of the coefficient matrix in the process of numerical calculation is relatively large; however, the expanded splitting mixed finite volume element scheme can first solve the coupling system of two equations to get the numerical solutions of the variables λ and σ, and then solve the third equation to get the numerical solution of the variable u, thus, greatly reduce the size of linear equations and the computational time. Finally, some numerical results are given to illustrate the feasibility and the correctness of the theoretical results.
In Chapter7, a splitting expanded mixed finite volume element method for a class of parabolic type integro-differential equation is formulated. By using transfer operator γh and introducing auxiliary variables λ(x,t)=-(?)u(x,t) and σ(x, t)=-(a(x)(?)u(x,t)+∫0tk(x, t,τ)(?)u(x, τ)dτ), both the semidiscrete and backward Euler fully discrete splitting expanded mixed finite volume element schemes are constructed. In the fully discrete scheme, left rectangle quadrature rule and backward Euler scheme are applied to getting the discretization of the integral term and the time derivative term, respectively. The optimal error estimates for two schemes are derived by introducing the Volterra type expanded mixed finite volume projection and using the properties of the transfer operator. Finally, a numerical example is given to confirm the theoretical results.
引文
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