Grid-independent Construction of Multistep Methods

A new polynomial formulation of variable step size linear multistep methods is presented, where each k-step method is characterized by a fixed set of k− 1 or k parameters. This construction includes all methods of maximal order (p = k for stiff, and p = k+1 for nonstiff problems). Supporting time step adaptivity by construction, the new formulation is not based on extending classical fixed step size methods; instead classical methods are obtained as fixed step size restrictions within a unified framework. The methods are implemented in Matlab, with local error estimation and a wide range of step size controllers. This provides a platform for investigating and comparing different multistep method in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software, although variable order has not yet been included. Mathematics subject classification: 65L06, 65L05, 65L80