High Accuracy Numerical Methods for Ordinary Differential Equations with Discontinuous Right-hand Side

Abstract

Ordinary Differential Equations (ODEs) with discontinuous right-hand side, where the discontinuities occur in the state variables arise in a number of applications. The appropriate formulation of such a problem (Filippov (1960)) is to replace the ODE x$\sp\prime$ = by a differential inclusion x$\sp\prime F(x)$ where F is a set-valued function with nonempty compact convex values and is upper semi-continuous. Solutions exist, although they are not necessarily unique. Numerical methods to compute solutions of differential inclusions have already been developed and convergence results proven by Taubert (1976, 1981), Elliott (1985) and Niepage & Wendt (1985). However, these methods are no better than 1st order accurate, and 1st order accuracy holds only with additional assumptions. Further, unless solutions are unique, these methods only guarantee that a sub-sequence of the numerical solutions converge. It is assumed that the structure of the discontinuous ODE is as follows: there is a collection of disjoint open sets $R\sb{i}$ with dense union in R$\sp{n}$, such that $\in R\sb{i}$ implies that f(x) = $f\sb{i}(x)$ with $f\sb{i}$ smooth, and the $R\sb{i}$ can be described by functions: $R\sb{i}$ = $\{x \vert h\sb{i}(x) < h\sb{j}(x)$, j $\not= i\}$. The active set at is $I(x) = \{i \vert x \in \=R\sb{i}\}$. The Filippov formulation becomes $x\sp\prime \in$ co$\{f\sb{i}(x) \vert i \in I(x)\}$. We suppose that I(x(t)) is a piecewise constant function. Where I(x(t)) is constant, we can select elements of the above differential inclusion that keep the active set constant. At a switching point the set of possible new values for I(x(t)) are determined by means of a Linear Complementarity Problem. Given a new active set from this set, we can continue by solving a new smooth ODE to maintain the new active set, until another switching point occurs. A complete algorithm using this approach has been developed and convergence results proven. In particular, if a particular solution has only finitely many switching points and some non-degeneracy conditions are satisfied, then by suitable choices of new active sets, a sequence of numerical solutions can be generated that converge to the given solution with the same order of accuracy the smooth ODE solver used. This algorithm has been implemented and numerical results obtained. A decomposition extension to the main algorithm is described and convergence results are given for this extension. This extension has been implemented and numerical results obtained for a friction problem with multiple friction surfaces.