L(\(\theta \))-stable peer methods with reused stages for advection–diffusion-reaction problems

Abstract

This paper addresses the efficient numerical solution of large and stiff initial value problems (IVPs) arising from the space discretization of systems of nonlinear advection–diffusion-reaction partial differential equations (PDEs). To this end, we introduce a new family of linearly implicit two-step peer methods that leverage specialized preconditioners and exploit the reuse of previously computed stages. The proposed methods are constructed to ensure strong stability properties, specifically, L-stability or L(\(\theta \))-stability with \(\theta \) approaching 90\(^{\circ }\), while maintaining low error constants. Compared to recently developed linearly implicit peer schemes, the new methods significantly reduce both the number of required function evaluations and linear systems to solve at each time step, also resulting in notably diminished error constants. Numerical experiments on nonlinear advection–diffusion-reaction problems testify to the efficiency of the new methods and confirm their accuracy and stability properties.