New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient

We examine the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. Such equations can be derived from the Fokker-Planck equation and are essential for understanding the diffusion mechanisms, e.g., in carbon nanotubes. First, we construct new, nontrivial analytical solutions with the classical self-similar Ansatz in one space dimension. Then we apply 14 different explicit numerical time integration methods, most of which are recently introduced unconditionally stable schemes, to reproduce the analytical solution. The test results show that the best algorithms, especially the leapfrog-hopscotch, are very efficient and severely outperform the conventional Runge-Kutta methods. Our results may attract attention in the community who develops multi-physics engineering software.

Automated summary

In this study, we investigate the one-dimensional transient diffusion equation with a space-dependent diffusion coefficient. The research is important for understanding diffusion mechanisms and presents new analytical solutions as well as efficient numerical methods. The results may be of interest to the community developing multi-physics engineering software.