Time‐splitting procedures for the solution of the two‐dimensional transport equation

Abstract

Purpose

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and as well as in many other branches of science and engineering. So it is essential to approximate the solution of these kinds of partial differential equations numerically in order to investigate the prediction of the mathematical models, as the exact solutions are usually unavailable.

Design/methodology/approach

The difficulties arising in numerical solutions of the transport equation are well known. Hence, the study of transport equation continues to be an active field of research. A number of mathematicians have developed the method of time‐splitting to divide complicated time‐dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time‐step. For example, they split large multi‐dimensional equations into a number of simpler one‐dimensional equations each solved separately over a fraction of the time‐step in the so‐called locally one‐dimensional (LOD) method. In the same way, the time‐splitting process can be used to subdivide an equation incorporating several physical processes into a number of simpler equations involving individual physical processes. Thus, instead of applying the one‐dimensional advection‐diffusion equation over one time‐step, it may be split into the pure advection equation and the pure diffusion equation each to be applied over half a time‐step. Known accurate computational procedures of solving the simpler diffusion and advection equations may then be used to solve the advection‐diffusion problem.

Findings

In this paper, several different computational LOD procedures were developed and discussed for solving the two‐dimensional transport equation. These schemes are based on the time‐splitting finite difference approximations.

Practical implications

The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared.

Originality/value

A comparison of calculations with the results of the conventional finite difference techniques demonstrates the good accuracy of the proposed approach.

Citation

Dehghan, M. (2007), "Time‐splitting procedures for the solution of the two‐dimensional transport equation", Kybernetes, Vol. 36 No. 5/6, pp. 791-805. https://doi.org/10.1108/03684920710749857