Explicit and implicit error inhibiting schemes with post-processing

TL;DR

This paper develops a new class of error inhibiting numerical schemes for differential equations that, through post-processing, achieve solutions two orders more accurate than traditional methods based on local truncation error analysis.

Contribution

It extends existing error inhibiting frameworks to include methods with exact leading error computation, enabling post-processing to significantly improve solution accuracy.

Findings

Methods achieve two orders higher accuracy after post-processing.

New explicit and implicit schemes are constructed and tested.

The schemes outperform traditional methods in various differential equations.

Abstract

Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global error that is completely determined by analysis of the local truncation error. In prior work in we investigated the interplay between the local truncation error and the global error to construct {\em error inhibiting schemes} that control the accumulation of the local truncation error over time, resulting in a global error that is one order higher than expected from the local truncation error. In this work we extend our error inhibiting framework to include a broader class of time-discretization methods that allows an exact computation of the leading error term, which can then be post-processed to obtain a solution that is two orders higher than expected from truncation error analysis. We define sufficient conditions that result in a desired form of the error and describe the construction of the post-processor. A number of new explicit and implicit methods that have this property are given and tested on a variety of ordinary and partial differential equation. We show that these methods provide a solution that is two orders higher than expected from truncation error analysis alone.