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.     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

三维多面体网格上扩散方程的保正格式

 针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性.     

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.     

Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ ‐method in time for solving time‐dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are nonobtuse, and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay‐type conditions. Numerical results are presented to verify the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the maximum principle and its application to diffusion equations

In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi‐discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h²) and O(h⁴) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h²) uniformly convergent discrete scheme or by O(h⁴) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A parallel finite volume scheme preserving positivity for diffusion equation on distorted meshes

Parallel domain decomposition methods are natural and efficient for solving the implicity schemes of diffusion equations on massive parallel computer systems. A finite volume scheme preserving positivity is essential for getting accurate numerical solutions of diffusion equations and ensuring the numerical solutions with physical meaning. We call their combination as a parallel finite volume scheme preserving positivity, and construct such a scheme for diffusion equation on distorted meshes. The basic procedure of constructing the parallel finite volume scheme is based on the domain decomposition method with the prediction‐correction technique at the interface of subdomains: First, we predict the values on each inner interface of subdomains partitioned by the domain decomposition. Second, we compute the values in each subdomain using a finite volume scheme preserving positivity. Third, we correct the values on each inner interface using the finite volume scheme preserving positivity. The resulting scheme has intrinsic parallelism, and needs only local communication among neighboring processors. Numerical results are presented to show the performance of our schemes, such as accuracy, stability, positivity, and parallel speedup.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2159–2178, 2017     

A study on a second order finite difference scheme for fractional advection–diffusion equations

Second order finite difference schemes for fractional advection–diffusion equations are considered in this paper. We note that, when studying these schemes, advection terms with coefficients having the same sign as those of diffusion terms need additional estimates. In this paper, by comparing generating functions of the corresponding discretization matrices, we find that sufficiently strong diffusion can dominate the effects of advection. As a result, convergence and stability of schemes are obtained in this situation.     

Convergence of a finite volume scheme for coagulation-fragmentation equations

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Convergence of a finite-volume scheme for the drift-diffusion equations in 1D

We construct an approximate solution to one-dimensional nonlineardrift–diffusion equations by a finite-volume scheme. Theconvergence of the scheme is proved, which yields the globalexistence of solutions to the equations. This result is valideven in the presence of vacuum state: the densities of freecarriers of charge vanish for some times and positions, whichimplies that the continuity equations are degenerate parabolicequations. Finally, numerical simulations are performed by thefinite-volume scheme.     

Stability of a colocated finite volume scheme for the Navier‐Stokes equations

The aim of this article is to describe a colocated finite volume approximation of the incompressible Navier‐Stokes equation and study its stability. One of the advantages of colocated finite volume space discretizations over staggered space discretizations is that all the variables share the same location; hence, the possibility to more easily use complex geometries and hierarchical decompositions of the unknowns. The time discretization used in the scheme studied here is a projection method. First, we give the full discretization of the incompressible Navier‐Stokes equations, then, we state the stability result and prove it following the methods of Marion and Temam. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Discrete maximum principle for the finite element solution of linear non-stationary diffusion–reaction problems

The maximum principle is one of the basic characteristic properties of solutions of second order partial differential equations of parabolic (and elliptic) types. The preservation of this property for solutions of corresponding discretized problems is a very natural requirement in reliable and meaningful numerical modelling of various real-life phenomena (heat conduction, air pollution, etc.). In the present paper we analyse a full discretization of a quite general class of linear parabolic equations and present sufficient conditions for the validity of a discrete analogue of the maximum principle in the case when bilinear finite elements are used for discretization in space.     

A mixed finite volume scheme for anisotropic diffusion problems on any grid

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.     

Superconvergence and gradient recovery for a finite volume element method for solving convection‐diffusion equations

We study the superconvergence of the finite volume element (FVE) method for solving convection‐diffusion equations using bilinear trial functions. We first establish a superclose weak estimate for the bilinear form of FVE method. Based on this estimate, we obtain the H¹‐superconvergence result: . Then, we present a gradient recovery formula and prove that the recovery gradient possesses the ‐order superconvergence. Moreover, an asymptotically exact a posteriori error estimate is also given for the gradient error of FVE solution.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1152–1168, 2014     

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5- and 7-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.     

Stability of a combined finite element ‐ finite volume discretization of convection‐diffusion equations

We consider a time‐dependent and a stationary convection‐diffusion equation. These equations are approximated by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix‐Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the nonstationary case, we use an implicit Euler approach for time discretization. This scheme is shown to be L²‐stable uniformly with respect to the diffusion coefficient. In addition, it turns out that stability is unconditional in the time‐dependent case. These results hold if the underlying grid satisfies a condition that is fulfilled, for example, by some structured meshes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 402–424, 2012     

An error estimate for finite volume methods for the Stokes equations on equilateral triangular meshes

We give here an error estimate for a finite volume discretization of the Stokes equations in two space dimensions on equilateral triangular meshes. This work was initiated by an analogous result presented by Alami‐Idrissi and Atounti for general triangular meshes. However, in this latter article, the result is not actually proven. We state here the restricting assumptions (namely equilateral triangles) under which the error estimate holds, using the tools which were introduced by Eymard, Gallouet and Herbin and used by Alami‐Idrissi and Atounti. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Error estimates for a finite element-finite volume discretization of convection-diffusion equations

We consider a time-dependent linear convection-diffusion equation. This equation is approximated by a combined finite element-finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements, and the convection term by upwind barycentric finite volumes on a triangular grid. An implicit Euler approach is used for time discretization. It is shown that the error associated with this scheme, measured by a discrete L^∞-L²- and L²-H¹-norm, respectively, decays linearly with the mesh size and the time step. This result holds without any link between mesh size and time step. The dependence of the corresponding error bound on the diffusion coefficient is completely explicit.