A fast compact exponential time differencing Runge-Kutta method for time-dependent advection-diffusion-reaction equations

A fast and accurate exponential Runge-Kutta method for a class of time-dependent advection-diffusion-reaction equations is developed. To discretise the convection term, a modified upwind difference scheme is used. This allows to avoid numerical oscillation and achieve second order spatial accuracy. The method demonstrates good stability and numerical examples show the applicability of the method to advection-diffusion-reaction problems with stiff nonlinearities.