Linearly Implicit Multistep Methods for Time Integration

TL;DR

This paper introduces linearly implicit multistep methods (LIMM) for time integration that require only solving linear systems per step, offering improved stability and performance over traditional BDF methods in large simulations.

Contribution

The paper presents a new family of LIMM methods with order up to five, including stability theory, design considerations, and practical implementation details.

Findings

LIMM methods require only one linear system solve per timestep.

LIMM methods have larger stability regions compared to BDF methods.

Numerical tests show performance improvements over BDF implementations.

Abstract

Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on timestep size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new LIMM methods shows measurable performance improvement over a similar BDF implementation.