Preprint series: 06-13 , Reports on Numerical Mathematics
The paper is published: Submitted for publication.
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 at least competitive with the linearly
implicit one-step methods provided in Kardos.
Keywords: Rosenbrock method, Peer method, Two-step method, Time-dependent PDE, Finite element software, Kardos
Upload: 2006-11-07