Diffusion is one of the simplest non-equilibrium processes. It describes the transport of heat and the time evolution of differences in substance concentrations. In this chapter, the one-dimensional diffusion equation is semi-discretized with finite differences. The time integration is performed with three different Euler methods. The explicit Euler method is conditionally stable only for small Courant number, which makes very small time steps necessary. The fully implicit method is unconditionally stable but its dispersion deviates largely from the exact expression. The Crank-Nicolson method is also unconditionally stable. However, it is more accurate and its dispersion relation is closer to the exact one. Extension to more than one dimension is easily possible, but the numerical effort increases drastically as there is no formulation involving simple tridiagonal matrices like in one dimension. The split operator approximation uses the one-dimensional method independently for each dimension. It is very efficient with almost no loss in accuracy. In a computer experiment the different schemes are compared for diffusion in two dimensions.
KeywordsDispersion Relation Diffusion Equation Matrix Notation Implicit Method Small Time Step
- 52.Y.A. Cengel, Heat transfer—A Practical Approach, 2nd edn. (McGraw-Hill, New York, 2003), p. 26. ISBN 0072458933, 9780072458930 Google Scholar
- 82.A. Fick, Philos. Mag. 10, 30 (1855) Google Scholar
- 95.J. Fourier, The Analytical Theory of Heat (Cambridge University Press, Cambridge, 1878), reissued by Cambridge University Press, 2009. ISBN 978-1-108-00178-6 Google Scholar