A comparison of one-step and two-step W-methods and peer methods with approximate matrix factorization

Abstract

One- and Two-step W-methods and peer methods are considered combined with an Approximate Matrix Factorization (AMF) for the numerical solution of large systems of stiff Ordinary Differential Equations (ODEs) coming from the spatial discretization of parabolic Partial Differential Equations (PDEs) of convection-diffusion-reaction type. Although one-step AMF W-methods do not require starting values and have larger stability regions, they may suffer from order reduction for such PDE problems when timedependent Dirichlet boundary conditions are imposed. Here by numerical tests it is illustrated that two-step AMF W-methods and AMF peer methods avoid this order reduction (in stiff sense, i.e. for a fixed spatial grid) due to their high stage order. On the other hand, while AMF W-methods are linearly implicit, AMF peer methods require the solution of nonlinear equations in order to compute an advancing solution. In this case, the predictor considered for the internal stages influences both stability and accuracy. However, with only one Newton iteration, for a given number of stages, the three classes of methods have a similar computational cost.