High order linearly implicit methods for semilinear evolution PDEs

Abstract: This paper considers the numerical integration of semilinear evolution PDEs using the high order linearly implicit methods developped in a previous paper in the ODE setting. These methods use a collocation Runge--Kutta method as a basis, and additional variables that are updated explicitly and make the implicit part of the collocation Runge--Kutta method only linearly implicit. In this paper, we introduce several notions of stability for the underlying Runge--Kutta methods as well as for the explicit step on the additional variables necessary to fit the context of evolution PDE. We prove a main theorem about the high order of convergence of these linearly implicit methods in this PDE setting, using the stability hypotheses introduced before. We use nonlinear Schr\''odinger equations and heat equations as main examples but our results extend beyond these two classes of evolution PDEs. We illustrate our main result numerically in dimensions 1 and 2, and we compare the efficiency of the linearly implicit methods with other methods from the litterature. We also illustrate numerically the necessity of the stability conditions of our main result.