Numerical fixed grid methods for differential inclusions

Abstract

Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.

Confinement Strategies in a Model for the Interaction between Individuals and a Continuum

“…This method is based on fixed discretizations in space and time, characterized by the space mesh size Δx and by the time step Δt, and using the classical Euler method to deal with the ordinary differential equation. We refer to [4] for further details and for a justification of this algorithm.…”

“…We consider here only cases with n = 2 and use the numerical method described in [4]. This method is based on fixed discretizations in space and time, characterized by the space mesh size Δx and by the time step Δt, and using the classical Euler method to deal with the ordinary differential equation.…”

Efficient computation of the bounds of continuous time imprecise Markov chains

“…For a survey on discretisation methods for the related problem of estimating the reachable sets for differential inclusions, see [7], or for some more recent applications [3,8].…”

Numerical solutions to noisy systems

“…For example, ellipsoidal calculus was used by Valyi and Kurzhanski [10], Lohner-type algorithm by Zgliczynski and Kapela [16], grid methods by Puri, Varaiya, and Borkar [11], also by Beyn and Rieger [3], discrete approximations by Dontchev and Farkhi [6], also by Grammel [8]. However, these algorithms either do not give rigorous over-approximations, or are approximations of low-order (Euler approximations with a first-order singlestep truncation error).…”

“…However, higher order discretization of a state space highly effects efficiency of the algorithm. It was noted in [3] that if one is trying to obtain higher order error estimates on the solution set of differential inclusions then grid methods should be avoided.…”