Efficient linearly implicit methods for nonlinear multidimensional parabolic problems

In this paper we present and analyze new methods to integrate multidimensional parabolic problems with nonlinear reaction terms. We consider a first standard spatial semidiscretization stage obtaining a family of Stiff nonlinear Initial Value Problems. The totally discrete schemes are obtained by numerical integration in time of such problems, using new Additive Runge–Kutta schemes. We show that the resulting algorithms, which are only linearly implicit, reach unconditional convergence, if the Additive methods used have suitable properties of linear absolute stability. Besides, they have a computational cost per time step with the same order as the explicit methods. Finally, three numerical experiences are shown in order to illustrate the behavior of our methods.