New Analytical Results and Numerical Schemes for Irregular Diffusion Processes

Abstract

We examine the transient diffusion equation with time- and space-dependent diffusion coefficients in 1D. Such transport equations can be easily derived from the Fokker-Planck equation and are essential to understand the diffusion mechanisms in general, e.g. in carbon nanotubes. With the help of the classical self-similar Ansatz we give new, nontrivial analytical solutions. Then we reproduce these by 16 explicit numerical time integration methods, 11 of which are recent and unconditionally stable. The results show that some of the algorithms, e.g. the leapfrog-hopscotch method, are very efficient and can outperform the standard FTCS method.