Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side
type: Item
instance of: scholarly article
title: Numerical solution of discontinuous differential systems: Approaching the discontinuity surface from one side
zbMATH Open document ID: 1268.65093
author: Luca Dieci
author: Luciano Lopez
published in: Applied Numerical Mathematics
publication date: 2013-04-18T00:00:00Z
review text: This paper deals with the numerical solution of initial value problems (IVPs) (1) \( x'(t) = f(x(t)), ~x(0)=x_0 \in {\mathbb R}^d\) where \(f\) is a discontinuous vector field defined by \( f(x)= f_1(x)\) if \( x \in S_1 = \{ x ; h(x)>0\} \) and \( f(x)=f_2(x)\) if \( x \in S_2 = \{ x ; h(x)< 0\} \) with a known discontinuity hypersurface in the \(d\)-dimensional space defined by \( \{x \in R^d; h(x)=0 \}\) where \( f_1\) may not be defined in \(S_2\) and \(f_2\) in \(S_1\). After some comments on the extension of these discontinuous systems into Filippov convex differential inclusions that allows one to define solutions that may slide along the discontinuity hypersurface or cross it, the authors consider the so-called one-sided explicit Runge-Kutta methods for solving one-sided solutions of IVPs (1). Such methods permit to define numerically a solution of (1) up to the point that hits the discontinuous hypersurface and are defined by adjusting the stepsize so that the vector field is computed only in the same side of the hypersurface. Some particular Runge-Kutta methods up to fourth order are considered and the numerical experiments with two elementary test problems are presented to show the performance of the methods.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L50
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 6155392
zbMATH Keywords: initial value problems
zbMATH Keywords: discontinous vector fields
zbMATH Keywords: explicit Runge-Kutta methods
zbMATH Keywords: detection of singularities
zbMATH Keywords: stepsize adjusting
zbMATH Keywords: Filippov convex differential inclusions
zbMATH Keywords: one-sided solution
zbMATH Keywords: numerical experiments
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: j.apnum.2011.08.010
OpenAlex ID: W2119268509
DOI: 10.1016/J.APNUM.2011.08.010
description: scientific article; zbMATH DE number 6155392