扩散方程保正的有限体积格式

在大变形网格上数值求解多介质扩散方程时, 如何构造具有保正性的扩散格式一直是人们关注的难题. 本文将简要综述与保正性相关的扩散格式的研究历史, 并为解决这一难题提出新的设计途径,构造出新的具有较高精度的单元中心型守恒保正格式, 它们可兼顾网格几何变形和物理量变化. 本文将给出数值实验结果, 验证新格式在变形的网格上保持非负性.     

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes. The vertex unknowns are treated as primary ones for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is not convex in general, leading to a fixed stencil of the flux approximation and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.     

Parallel algorithm for the solutions of PDEs in linux clustered workstations

In this paper we propose parallel algorithm for the solution of partial differential equations over a rectangular domain using the Crank–Nicholson method by cooperation with the DuFort–Frankel method and apply it on a model problem, namely, the heat conduction equation. One of the well known parallel techniques in solving partial differential equations in cluster computing environment is the domain decomposition technique. Using this technique, the whole domain is decomposed into subdomains, each of them has its own boundaries that are called the interface points. Parallelization is realized by approximating interface values using the unconditionally stable DuFort–Frankel explicit scheme, and these values serve as Neumann boundary conditions for the Crank–Nicholson implicit scheme in the subdomains. The numerical results show that our algorithm is more accurate than the algorithm based on the forward explicit method to approximate the values of the interface points, especially, when we use a small number of time steps. Moreover, these numerical results show that increasing the number of processors which are used in the cluster, yields an increase in the algorithm speedup.     

扩散方程一种无条件稳定的保正并行有限差分方法

本文针对扩散方程提出了一种保正的并行差分格式,并且这个格式为无条件稳定的.我们在每个时间层将计算区域分成许多个子区域以便于实施并行计算.格式构造中首先我们使用前两个时间层的计算结果在分区界面处通过一种非线性的保正外插来预估子区域界面值.然后在每个子区域内部使用经典的全隐格式进行计算.最后在界面处使用全隐格式进行校正（本质上这一步计算是显式计算）.我们给出了一维与二维情形下的保正并行差分格式,并相应的给出了无条件稳定性证明.数值实验显示此并行格式具有二阶数值精度,而且无条件稳定性与保正性也均在数值实验中得到验证.     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations

Based on two-grid discretizations, some local and parallel finite element algorithms for the d-dimensional (d = 2,3) transient Stokes equations are proposed and analyzed. Both semi- and fully discrete schemes are considered. With backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Stokes equations using a coarse grid on the entire domain, then correct the resulted residue using a finer grid on overlapped subdomains by some local and parallel procedures at each time step. By the technical tool of local a priori estimate for the fully discrete finite element solution, errors of the corresponding solutions from these algorithms are estimated. Some numerical results are also given which show that the algorithms are highly efficient.     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

Studies of an interface relaxation domain decomposition technique using finite elements on a parallel computer

Studies are presented for an interface relaxation domain decomposition technique using finite elements on an iPSC/2 D5 Hypercube Concurrent computer. The general type of problem to be solved is one governed by a partial differential equation. The application of the approach, however, will be extended to a free boundary value problem by appropriate modification of the numerical scheme. Using the domain decomposition technique, the computation domain is subdivided into several subdomains. In addition, on the interfaces between two adjacent subdomains are imposed a continuity condition on one side and an equilibrium condition on the other side. Successive overrelaxation iterative processes are then carried out in all subdomains with a relaxation process imposed on the interfaces. With this domain decomposition technique, the problem can be solved parallelly until convergence is reached both in the interiors and on the interfaces of all subdomains. Moreover, the formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee load balancing. Furthermore, it is shown, through numerical experiments performed on an example problem of free surface seepage through a porous dam, how the values of the relaxation parameters, the choice of imposed boundary conditions, and the number of subdomains (i.e., the number of processors used) affect the solution convergence in this parallel computing environment. © 1993 John Wiley & Sons, Inc.     

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms. In this work, we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique, modified upwind differences with explicitimplicit coupling, the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy. Moreover, for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictorcorrector or stabilized schemes. These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.     

多边形网格上扩散方程新的单调格式

 本文在星形多边形网格上, 构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是, 在网格边的法向流离散模板中包含当前边上的点, 在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量, 这样既可适应网格几何大变形, 同时又兼顾了当前网格边上物理量的变化. 在光滑解情形证明了离散法向流的相容性.对于具有强各向异性、非均匀张量扩散系数的扩散方程, 证明了新格式是单调的, 即格式可以保持解析解的正性. 数值结果表明在扭曲网格上, 所构造的格式是局部守恒和保正的, 对光滑解有高于一阶的精度, 并且, 针对非平衡辐射限流扩散问题, 数值结果验证了新格式在计算效率和守恒精度上优于九点格式.     

A PRACTICAL PARALLEL DIFFERENCE SCHEME FOR PARABOLIC EQUATIONS

A practical parallel difference scheme for parabolic equations is constructed as follows: to decompose the domain Ω into some overlapping subdomains, take flux of the last time layer as Neumann boundary conditions for the time layer on inner boundary points of subdomains, solve it with the fully implicit scheme on each subdomain, then take correspondent values of its neighbor subdomains as its values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points. The scheme is unconditionally convergent. Though its truncation error is O(τ h), the convergent order for the solution can be improved to O(τ h2).     

A new cement to glue non-conforming grids with Robin interface conditions: The finite volume case

Summary Robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in the convergence. Non conforming grids make the grid generation much easier and faster since it is then a parallel task. The goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of Robin interface conditions with non-matching grids. We consider both a symmetric definite positive operator and the convection-diffusion equation discretized by finite volume schemes. Numerical results are shown. Received December 22, 1999 / Revised version received December 21, 2000 / Published online December 18, 2001 Correspondence to: F. Nataf     

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions

In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L²‐norm. Numerical experiments are presented to illustrate the theoretical analysis.     

结构三角网上抛物方程的有限差分区域分解算法

1引言对于大型科学与工程计算问题,并行计算是必需的．构造高效率的数值并行方法一直是人们关心的问题,并且已有了大量的研究．在三层交替计算方法的研究中出现了许多既具有明显并行性又绝对稳定的差分格式(见[1]-[5])．在只涉及两个时间层的算法研究中,Dawson等人(见[6])首先发展了求解一维热传导方程的区域分解算法,并将其推广到     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

一类无结构三角网上抛物方程的有限差分区域分解算法

本文讨论了一类在无结构三角网上数值求解二维热传导方程的有限差分区域分解算法．在这个算法中,将通过引进两类不同类型的内界点,将求解区域分裂成若干子区域．一旦内界点处的值被计算出来,其余子区域上的计算可完全并行．本文得到了稳定性条件和最大模误差估计,它表明我们的格式有令人满意的稳定性和较高的收敛阶．     

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.     

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.     

Iterative methods for a forward-backward heat equation in two-dimension

A finite difference method is introduced to solve the forward-backward heat equation in two space dimensions. In this procedure, the backward and forward difference scheme in two subdomains and a coarse-mesh second-order central difference scheme at the middle interface are used. Maximum norm error estimate for the procedure is derived. Then an iterative method based on domain decomposition is presented for the numerical scheme and the convergence of the given method is established. Then numerical experiments are presented to support the theoretical analysis.