A Software Platform for Adaptive High Order Multistep Methods

Author:

Arévalo Carmen1,Jonsson-Glans Erik2,Olander Josefine2,Soto Monica Selva3,Söderlind Gustaf4

Affiliation:

1. Center for Mathematical Sciences, Lund Institute of Technology, Lund, Sweden

2. HiQ Ace AB, Linköping, Lund, Sweden

3. Department of Mathematical Engineering, University of Concepción, Chile

4. Center for Mathematical Sciences, Lund University, Lund, Sweden

Abstract

We present a software package, M odes , offering h -adaptive and p -adaptive linear multistep methods for first order initial value problems in ordinary differential equations. The implementation is based on a new parametric, grid-independent representation of multistep methods [Arévalo and Söderlind 2017]. Parameters are supplied for over 60 methods. For nonstiff problems, all maximal order methods ( p = k for explicit and p = k +1 for implicit methods) are supported. For stiff computation, implicit methods of order p = k are included. A collection of step-size controllers based on digital filters is provided, generating smooth step-size sequences offering improved computational stability. Controllers may be selected to match method and problem classes. A new system for automatic order control is also provided for designated families of multistep methods, offering simultaneous h - and p -adaptivity. Implemented as a M atlab toolbox, the software covers high order computations with linear multistep methods within a unified, generic framework. Computational experiments show that the new software is competitive and offers qualitative improvements. M odes is available for downloading and is primarily intended as a platform for developing a new generation of state-of-the-art multistep solvers, as well as for true ceteris paribus evaluation of algorithmic components. This also enables method comparisons within a single implementation environment.

Publisher

Association for Computing Machinery (ACM)

Subject

Applied Mathematics,Software

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