A finite volume method for one-dimensional reaction–diffusion problems

A truly conservative finite volume method for one-dimensional reaction–diffusion equations is presented. The method is based on piecewise second-degree polynomials approximations that result in linear fluxes, a symmetric mass matrix and a semi-discrete system of nonlinear ordinary differential equations in time. It is shown that the semi-discrete method is linearly stable provided that the problem is mathematically well-posed. It is also shown that implicit discretizations of the time variable are stable, whereas the explicit one is conditionally stable and has a smaller domain of linear stability than the standard explicit finite difference method. It is also illustrated that, upon lumping the mass matrix, the method coincides with a finite difference discretization of the original problem. A detailed formulation of the Dirichlet, Neumann and Robin boundary conditions is presented. Such a formulation makes use of control volumes which are here referred to as incomplete ones. Although, the finite volume technique presented here uses a node-centered formulation, it can be easily extended to cell-centered ones. The technique is also developed for multidimensional problems.