Linearly implicit Runge-Kutta methods and approximate matrix factorization

Linearly implicit Runge-Kutta methods are a class of suitable time integrators for initial value problems of ordinary differential systems whose right-hand side function can be written as the sum of a stiff linear part and a nonlinear term. Such systems arise for instance after spatial discretization of taxis-diffusion-reaction systems from mathematical biology. When approximate matrix factorization is used for efficiently solving the stage equations appearing in these methods, then the order of the methods is reduced to one. In this paper we analyse this fact and propose an appropriate and efficient correction to achieve order two while preserving the main stability properties of the underlying method. Numerical experiments with LIRK3 [Appl. Numer. Math. 37 (2001) 535] illustrating the theory are provided. In the case of taxis-diffusion-reaction systems, the corrected method compares well with other suitable schemes.