Estimating Global Errors in Time Stepping

TL;DR

This paper proposes new time-stepping methods with integrated global error estimators, enabling more efficient and accurate solutions of differential equations through overlapped computations and general linear methods.

Contribution

It introduces a novel approach to global error estimation that generalizes classical schemes and allows for overlapped internal computations.

Findings

Effective global error estimation demonstrated on examples

New explicit self-starting schemes similar to Runge-Kutta methods

Theoretical framework for general linear methods with error control

Abstract

This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. We present a few explicit self-starting schemes akin to Runge-Kutta methods with global error estimation and illustrate the theoretical considerations on several examples.