High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

The main purpose of the paper is to show how to use implicit–explicit Runge–Kutta methods in a much more general context than usually found in the literature, obtainingvery effective schemes for a large class of problems. This approach gives a great flexibility,and allows, in many cases the construction of simple linearly implicit schemes without anyNewton’s iteration. This is obtained by identifying the (possibly linear) dependence on theunknown of the system which generates the stiffness. Only the stiff dependence is treatedimplicitly, thenmaking the wholemethod much simpler than fully implicit ones. The resultingschemes are denoted as semi-implicit R–K.We adopt several semi-implicit R–K methods upto order three.We illustrate the effectiveness of the new approach with many applications toreaction–diffusion, convection diffusion and nonlinear diffusion system of equations