Stability analysis of Crank–Nicolson and Euler schemes for time-dependent diffusion equations

Abstract

In this paper, we study the stability of the Crank–Nicolson and Euler schemes for time-dependent diffusion coefficient equations on a staggered grid with explicit and implicit approximations to the Dirichlet boundary conditions. Using the matrix representation for the numerical scheme and boundary conditions it is shown that for implicit boundary conditions the Crank–Nicolson scheme is unrestrictedly stable while it becomes conditionally stable for explicit boundary conditions. Numerical examples are provided illustrating this behavior. For the Euler schemes the results are similar to those for the constant coefficient case. The implicit Euler with implicit or explicit boundary conditions is unrestrictedly stable while the explicit Euler with explicit boundary conditions presents the usual stability restriction on the time step.