A new embedded variable stepsize, variable order family of low computational complexity

TL;DR

This paper introduces a family of implicit, embedded variable stepsize, variable order methods for solving ODEs that are computationally efficient and easy to implement, requiring only one BDF solve per step.

Contribution

The authors develop implicit and linearly implicit VSVO methods of orders two to four with minimal computational complexity, compatible with legacy code, and simple to adapt.

Findings

Methods require only one BDF solve per step.

Constructed methods match the complexity of BDF3.

Changing the order is straightforward without extra solves.

Abstract

Variable Stepsize Variable Order (VSVO) methods are the methods of choice to efficiently solve a wide range of ODEs with minimal work and assured accuracy. However, VSVO methods have limited impact in timestepping methods in complex applications due to their computational complexity and the difficulty to implement them in legacy code. We introduce a family of implicit, embedded, VSVO methods that require only one BDF solve at each time step followed by adding linear combinations of the solution at previous time levels. In particular, we construct implicit and linearly implicit VSVO methods of orders two, three and four with the same computational complexity as variable stepsize BDF3. The choice of changing the order of the method is simple and does not require additional solves of linear or nonlinear systems.