Finite difference methods for three kinds of reaction-diffusion equations with free boundaries

Abstract

This work considers to numerically solve three kinds of reaction-diffusion equations with free boundaries. First, the popular front-fixing method is used to transform the considered free boundary problems into fixed boundary problems. Then, by employing the finite difference method, numerical schemes with $\mathrm{M}$-matrices as their coefficient matrices are developed for these transformed fixed-boundary problems. Next, for the developed numerical schemes, we establish numerical theory involving positivity preservation, monotonicity preservation, and stability. It is noteworthy that, unlike some existing works that impose tight restrictions on the time step-size to discuss the numerical theory, the proposed numerical schemes require only a mild restriction. Finally, numerical examples are provided to test the developed numerical schemes and to confirm the theoretical results.