On the stability and convergence of an implicit logarithmic scheme for diffusion equations with nonlinear reaction

In this work, we investigate numerically a 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. The theoretical analysis of the numerical model is carried out also. In particular, we establish that the method is a consistent technique, we provide a priori bounds for the numerical solutions, and we prove the stability and the convergence of the scheme by applying a suitable Gronwall-type inequality. As one of the consequences of stability, we show not only that the solutions of the numerical model exist, but also that they are unique.