Exponential methods for one-dimensional reaction–diffusion equations

摘要

Non-iterative exponential time-linearization and iterative exponential quasilinearization techniques for one-dimensional reaction–diffusion equations based on the discretization of the time derivative, the freezing of the coefficients of the resulting linear ordinary differential equations and the piecewise analytical solution of these ordinary differential equations, are presented and applied to the Nagumo and Fisher equations. These techniques yield three-point finite difference expressions that depend in an exponential manner on either the diffusion, reaction and transient terms or the diffusion and reaction terms. It is shown that first-order accurate in time, exponential time-linearization and exponential quasilinearization methods which account for reaction and diffusion processes in the (ordinary) differential operator are more robust than standard second-order accurate techniques even for very large time steps and grid sizes, and yield results in excellent accord with those obtained by means of non-standard finite difference schemes and nodal integral formulations for the Nagumo and Fisher equations. Unlike non-standard finite difference methods, the exponential schemes presented in this paper do not require any knowledge of the exact solution of the differential equation and can be easily derived in a rather systematic manner. Compared with nodal integral formulations, the exponential techniques presented here only require the solution of second-order ordinary differential equations.