Discrete Maximum Principle (discrete + maximum_principle)

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Selected Abstracts


An upwind finite volume element method based on quadrilateral meshes for nonlinear convection-diffusion problems

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2009
Fu-Zheng Gao
Abstract Considering an upwind finite volume element method based on convex quadrilateral meshes for computing nonlinear convection-diffusion problems, some techniques, such as calculus of variations, commutating operator, and the theory of prior error estimates and techniques, are adopted. Discrete maximum principle and optimal-order error estimates in H1 norm for fully discrete method are derived to determine the errors in the approximate solution. Thus, the well-known problem [(Li et al., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Marcel Dekker, New York, 2000), p 365.] has been solved. Some numerical experiments show that the method is a very effective engineering computing method for avoiding numerical dispersion and nonphysical oscillations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009 [source]


Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2003
N. SukumarArticle first published online: 11 MAR 200
Abstract Voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest-point regions. The Sibson and the Laplace (non-Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two-fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well-known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive-definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley & Sons, Ltd. [source]


The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2002
H. Hoteit
Abstract The abundant literature of finite-element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial,boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time-dependent solutions for such problems during small time duration obtained by using a non-conforming mixed-hybrid finite-element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite-difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non-physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial,boundary value problem will be presented. One of the propositions given to avoid any non-physical oscillations is to use the mass-lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd. [source]


Consistency with continuity in conservative advection schemes for free-surface models

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2002
Edward S. Gross
Abstract The consistency of the discretization of the scalar advection equation with the discretization of the continuity equation is studied for conservative advection schemes coupled to three-dimensional flows with a free-surface. Consistency between the discretized free-surface equation and the discretized scalar transport equation is shown to be necessary for preservation of constants. In addition, this property is shown to hold for a general formulation of conservative schemes. A discrete maximum principle is proven for consistent upwind schemes. Various numerical examples in idealized and realistic test cases demonstrate the practical importance of the consistency with the discretization of the continuity equation. Copyright © 2002 John Wiley & Sons, Ltd. [source]