Structural and numerical analysis of an implicit logarithmic scheme for diffusion equations with nonlinear reaction

In this work, we investigate a numerical diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. Finally, an extensive set of numerical simulations is provided in order to illustrate the performance of the scheme. The results verify that the logarithmic scheme converges to the exact solution of the continuous problem, with first order of convergence in time and second order in space. Moreover, we provide some comparisons on the efficiency of the implicit method against an explicit logarithmic scheme of the literature. The results show that the present discretization is a more efficient technique.