Numerical Methods for Solving Convection-Diffusion Problems

TL;DR

This paper reviews numerical methods for solving convection-diffusion equations, focusing on discretization techniques, stability, and monotone approximations across various forms and applications.

Contribution

It provides a comprehensive analysis of discretization schemes, stability conditions, and monotone approximations for convection-diffusion problems in different formulations and computational frameworks.

Findings

Finite difference, finite volume, and finite element methods are analyzed.

Unconditionally stable explicit-implicit schemes are developed.

Stability conditions are derived for various formulations.

Abstract

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For different problems, a convection-diffusion equation may be be written in various forms. The most popular formulation of convective transport employs the divergent (conservative) form. In some cases, the nondivergent (characteristic) form seems to be preferable. The so-called skew-symmetric form of convective transport operators that is the half-sum of the operators in the divergent and nondivergent forms is of great interest in some applications. Here we discuss the basic classes of discretization in space: finite difference schemes on rectangular grids, approximations on general polyhedra (the finite volume method), and finite element procedures. The key properties of discrete operators are studied for convective and diffusive transport. We emphasize the problems of constructing approximations for convection and diffusion operators that satisfy the maximum principle at the discrete level --- they are called monotone approximations. Two- and three-level schemes are investigated for transient problems. Unconditionally stable explicit-implicit schemes are developed for convection-diffusion problems. Stability conditions are obtained both in finite-dimensional Hilbert spaces and in Banach spaces depending on the form in which the convection-diffusion equation is written.