An optimized second order numerical scheme applied to the non-linear Fisher’s reaction-diffusion equation

Abstract

Four one-parameter θ -family of simple and seemingly implicit finite difference schemes are proposed to obtain an accurate approximate solution for the non-linear Fisher model problem arising in population dynamics. The underlying methods are based upon the Saul’yev’s asymmetrical methods and the method of lagging. It is shown that, in general, the proposed schemes are consistent with the first and second-order accuracy in time and in space respectively. However, for two families, temporally and spatially second-order accuracy is achieved if one uses a special value of the weighting parameter. Hence, based on these families, a two-parameter (Θ,θ) -family of implicit finite difference approximation is introduced. By optimizing these parameters, we show that the latter scheme is second order accurate in both time and space. In fact, by selecting Θ,θ = 1/2, the stability of the method is established in the sense of von Neumann analysis. Both the accuracy and the convergence rates of the proposed discretized schemes are verified through numerical simulations and comparisons with available exact solutions and alternative existing numerical values are also made. Numerical results show the present methods and in particular the implicit approach are superior to the known methods in terms of accuracy.