Optimal Residuals and the Dahlquist Test Problem

TL;DR

This paper introduces a method to compute the optimal relative backward error for one-step numerical methods solving the Dahlquist test problem, providing new insights into stiff problem solutions using optimal control theory.

Contribution

It presents a novel approach combining optimal control theory with elementary methods to analyze residuals in the Dahlquist test problem, enhancing understanding of stiff numerical problems.

Findings

Optimal residuals can be computed explicitly for the Dahlquist problem.

The analysis offers new insights into the numerical treatment of stiff differential equations.

Elementary methods suffice for simple problems like the Dahlquist test case.

Abstract

We show how to compute the \emph{optimal relative backward error} for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces new insight into the numerical solution of stiff problems.