Multistep methods. A grid-independent formulation and its implementation (Carmen Arévalo, Lund)

14.03.2017 14:15

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, independent of the grid. This construction includes variable-step, implicit and explicit Adams methods of any order, and the Backward Differentiation Formulas (BDFs) up to order 6, as well as every other method of maximal order. Other types of special methods may also be constructed within this formulation, such as strong stability preserving (SSP) methods.

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. This new way of defining multistep methods has been used to develop a Matlab code for the solution of ODEs. With variable-order and a wide range of advanced step-size controllers, the solver provides a platform for investigating and comparing different multistep methods in realistic operational conditions. Computational experiments show that the new multistep method construction and implementation compares favorably to existing software.

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