One-Sided Event Location Techniques in the Numerical Solution of Discontinuous Differential Systems (original) (raw)
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Applied Numerical Mathematics
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.
Time Reparametrization and Event Location for Discontinuous Differential Algebraic Equations
Journal of scientific computing, 2024
In this paper, we consider numerical methods for the event location of differential algebraic equations. The event corresponds to cross a discontinuity surface, beyond which another differential algebraic equation holds. The methods are based on a particular change of the independent variable time, called time reparametrization or time transformation, reducing the equation to another equation where the event time is known in advance. From a numerical point of view, these methods never cross the discontinuity surface and reach it in a fixed number of steps. The methods works also for differential algebraic equations of index higher than one.
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2012
This work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numerical Analysis of the second author. The authors remember Donato as a generous teacher, always ready to discuss with his students, able to give them profound and interesting suggestions. Here, we present a survey of numerical methods for differential systems with discontinuous right hand side.
An event-driven method to simulate Filippov systems with accurate computing of sliding motions
ACM Transactions on Mathematical Software, 2008
This paper describes how to use smooth solvers for simulation of a class of piecewise smooth dynamical systems, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and require special treatment numerically. The introduced algorithms are based on an extension to Filippov's method to stabilize the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in Matlab and sufficient details are given to enable users to modify the code to run on arbitraray examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of a oil well drill-string.
Rosenbrock-type methods applied to discontinuous differential systems
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In this paper we will study the numerical solution of a discontinuous differential system by a Rosenbrock method. We will also focus on one-sided approach in the context of Rosenbrock schemes, and we will suggest a technique based on the use of continuous extension, in order to locate the event point, with an application to discontinuous singularly perturbed systems.
On Filippov and Utkin Sliding Solution of Discontinuous Systems
2009
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Optimal control of systems with discontinuous differential equations
Numerische Mathematik, 2009
In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with discontinuous right hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such right hand side. Notwithstanding the impressive development of non-smooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. First, we show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the step-size. This means that the strategy of "optimize the discretization" will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided the start and end points of the trajectory do not lie on the discontinuity, and that using Euler's method where the step size is "sufficiently small" in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing which involves Coulomb friction and slip showing that this approach is practical and can handle problems of considerable complexity.