Arbitrary high-order methods for one-sided direct event location in discontinuous differential problems with nonlinear event function

TL;DR

This paper develops high-order numerical methods for accurately locating one-sided events in discontinuous differential problems with nonlinear event functions, transforming the problem into a Poisson system and applying energy-conserving techniques.

Contribution

It introduces a novel approach that transforms the event location problem into a Poisson problem and adapts energy-conserving methods for high-order accuracy in nonlinear cases.

Findings

Effective transformation into Poisson problems.

Successful application of energy-conserving methods.

Numerical tests confirm theoretical accuracy.

Abstract

In this paper we are concerned with numerical methods for the one-sided event location in discontinuous differential problems, whose event function is nonlinear (in particular, of polynomial type). The original problem is transformed into an equivalent Poisson problem, which is effectively solved by suitably adapting a recently devised class of energy-conserving methods for Poisson systems. The actual implementation of the methods is fully discussed, with a particular emphasis to the problem at hand. Some numerical tests are reported, to assess the theoretical findings.