Algebraic-based nonstandard time-stepping schemes

TL;DR

This paper introduces algebraic-based nonstandard time-stepping schemes for differential equations, utilizing algebraic derivative estimation to enhance classical methods like Euler and Runge-Kutta, with potential filtering benefits.

Contribution

It proposes novel nonstandard time-stepping strategies using algebraic derivative estimation to improve classical numerical schemes for differential equations.

Findings

Algebraic estimation can replace finite differences in Euler methods.

Improved slope predictions in Runge-Kutta schemes using algebraic derivatives.

Potential filtering properties from algebraic derivative estimation.

Abstract

In this preliminary work, we present nonstandard time-stepping strategies to solve differential equations based on the algebraic estimation method applied to the estimation of time-derivative, which provides interesting properties of "internal" filtering. We consider firstly a classical finite difference method, like the explicit Euler method for which we study the possibility of using the algebraic estimation of derivatives instead of the usual finite difference to compute the numerical derivation. Then, we investigate how to use the algebraic estimation of derivatives in order to improve the slope predictions in RK-based schemes.