High-order linearly implicit two-step peer – finite element methods for time-dependent PDEs

Abstract

Linearly-implicit two-step peer methods are successfully applied in the numerical solution of ordinary differential and differential-algebraic equations. One of their strengths is that even high-order methods do not show order reduction in computations for stiff problems. With this property, peer methods commend themselves as time-stepping schemes in finite element calculations for time-dependent partial differential equations (PDEs). We have included a class of linearly-implicit two-step peer methods in the finite element software Kardos. There PDEs are solved following the Rothe method, i.e. first discretised in time, leading to linear elliptic problems in each stage of the peer method. We describe the construction of the methods and how they fit into the finite element framework. We also discuss the starting procedure of the two-step scheme and questions of local temporal error control. The implementation is tested for two-step peer methods of orders three to five on a selection of PDE test problems on fixed spatial grids. No order reduction is observed and the two-step methods are more efficient, at least competitive, in comparison with the linearly implicit one-step methods provided in Kardos.