Efficient A-stable peer two-step methods

A-stability is a desirable property for stiff integrators. However, it may be difficult to prove or implement in an automated search for multi-stage multi-step methods requiring eigenvalue computations on complex sets of parameters. In this paper we apply a purely algebraic criterion for A-stability of peer two-step methods requiring only the solution of algebraic equations and combine it with a new criterion for zero-stability of peer methods on general time grids. Both criteria lead to formulations solvable by standard numerical methods and are easily verifiable by anyone if sufficient data are provided. Several 4-stage methods of order 3 and 4 are constructed and compared with existing methods on a few standard test problems.