A software platform for adaptive high order multistep methods

TY - JOUR

T1 - A software platform for adaptive high order multistep methods

AU - Arévalo, Carmen

AU - Jonsson-Glans, Erik

AU - Olander, Josefine

AU - Soto, Monica Selva

AU - Söderlind, Gustaf

PY - 2020/4

Y1 - 2020/4

N2 - We present a software package, Modes, 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 Matlab 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. Modes 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.

AB - We present a software package, Modes, 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 Matlab 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. Modes 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.

KW - multistep methods

KW - ordinary differential equations

KW - Solver

KW - variable order

KW - variable step size

UR - http://www.scopus.com/inward/record.url?scp=85084748673&partnerID=8YFLogxK

U2 - 10.1145/3372159

DO - 10.1145/3372159

M3 - Article

AN - SCOPUS:85084748673

VL - 46

JO - ACM Transactions on Mathematical Software

JF - ACM Transactions on Mathematical Software

SN - 0098-3500

IS - 1

M1 - 2

ER -