Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations

Abstract

Abstract We solve two numerical experiments described by 2D nonconstant coefficient advection–diffusion equations with specified initial and boundary conditions. Three finite difference methods, namely Lax–Wendroff, Du-Fort–Frankel and a nonstandard finite difference scheme, are derived and used to solve the two problems, whereby only the first problem has an exact solution. Stability analysis is performed to obtain a range of values of the time step size at a fixed spatial step size. We obtain the rate of convergence in space when the three methods are used to solve Problem 1. Computational times of the three algorithms are computed for Problem 1. Results are displayed for the two problems using the three methods at times <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mn>1.0</m:mn> </m:math> T=1.0 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> <m:mo>=</m:mo> <m:mn>5.0</m:mn> </m:math> T=5.0 . The main novelty is the stability analysis, which is not straightforward as we are working with numerical methods discretising 2D nonconstant coefficient advection–diffusion equation where many parameters are involved. The second highlight is to determine the most efficient scheme from the three methods. Third, there are very few published studies on analysis and use of numerical methods to solve nonconstant coefficient advection–diffusion equations, and this is one of the very few rare articles treating such topics.