W-methods to stabilize standard explicit Runge–Kutta methods in the time integration of advection–diffusion–reaction PDEs

摘要

A technique to stabilize standard explicit Runge–Kutta methods by associating them with W-methods is proposed. The main point to get the associated family of W-methods for a given explicit Runge–Kutta method is to require commutativity for the coefficient matrices of the W-method in order to reduce the large number of order conditions that must be satisfied to get a pre-fixed order.Based on this idea, for any given explicit four-stage Runge–Kutta method of order four, two uniparametric families of third order W-methods are obtained. The free parameter can be used to increase the stability regions of the W-methods in case of d≥1 splittings in the derivative function when a von Neumann stability analysis is carried out. Additionally, it is possible to find L-stable ROW-methods (W-methods with exact Jacobian) for some specific values of the free parameter.The new family of W-methods is also equipped with the splitting provided by the Approximate Matrix Factorization (AMF), which converts a W-method into some kind of ADI-method (Alternating Direction Implicit method). The AMF-W-methods so obtained are mainly used to solve large time-dependent PDE systems (in 2D or 3D spatial variables) discretized in space by using finite differences or finite volumes. Some stability properties of the family of AMF-W-methods are also supplied for the case of d-splittings (d≥1).Numerical experiments in connection with the proposed AMF-W-methods on a few interesting stiff problems coming from PDE discretizations illustrate the stabilization approach in comparison with some relevant methods in the literature.