Analytical and Numerical Results for the Transient Diffusion Equation with Diffusion Coefficient Depending on Both Space and Time

Abstract

The time-dependent diffusion equation is studied, where the diffusion coefficient itself depends simultaneously on space and time. First, a family of novel, nontrivial analytical solutions is constructed in one space dimension with the classical self-similar Ansatz. Then, the analytical solution for two different sets of parameters is reproduced by 18 explicit numerical methods. Fourteen of these time integrators are recent unconditionally stable algorithms, which are often much more efficient than the mainstream explicit methods. Finally, the adaptive time-step version of some of these algorithms are created and tested versus widespread algorithms, such as the Runge–Kutta–Fehlberg solver.

Keywords

• diffusion
• heat conduction
• analytical solution
• explicit time integration
• unconditionally stable numerical methods
• adaptive step size controllers