A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side. (English) Zbl 1246.65111
Summary: 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. In particular, we will review methods where the discontinuities are detected by using an event function (so-called event driven methods) and methods where the discontinuities are located by controlling the local errors (so-called time-stepping methods). Particular attention will be devoted to discontinuous systems of Filippov’s type where sliding behavior on the discontinuity surface is allowed.
Here, we present a survey of numerical methods for differential systems with discontinuous right hand side. In particular, we will review methods where the discontinuities are detected by using an event function (so-called event driven methods) and methods where the discontinuities are located by controlling the local errors (so-called time-stepping methods). Particular attention will be devoted to discontinuous systems of Filippov’s type where sliding behavior on the discontinuity surface is allowed.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |
Keywords:
discontinuous ODEs; one-step and multistep methods; time-stepping methods; survey article; numerical examples; initial value problems (IVPs); ordinary differential equations (ODEs); differential systems; event driven methods; discontinuous systems of Filippov’s typeSoftware:
RODASReferences:
| [1] | Bartolini, G.; Parodi, F.; Utkin, V. I.A.; Zolezzi, T., The simplex method for nonlinear sliding mode control, Mathematical Problems in Engineering, 4, 461-487 (1999) · Zbl 0927.93020 |
| [2] | Chong, E. K.P.; Hui, S.; Zak, S. H., An analysis of a class of neural networks for solving linear programming problems, IEEE Transactions on Automatic Control, 44, 1995-2006 (1999) · Zbl 0957.90087 |
| [3] | Glazos, M. P.; Hui, S.; Zak, S. H., Sliding modes in solving convex programming problems, SIAM Journal on Control and Optimization, 36, 680-697 (1998) · Zbl 0911.93023 |
| [4] | Gouze, J. L.; Sari, T., A class of piecewise linear differential equations arsing in biological models, Dynamical Systems, 17, 299-319 (2002) · Zbl 1054.34013 |
| [5] | Johansson, K. H.; Barabanov, A. E.; Astrom, K. J., Limit cycles with chattering in relay feedback systems, IEEE Transactions on Automatic Control, 247, 1414-1423 (2002) · Zbl 1364.93341 |
| [6] | Johansson, K. H.; Rantzer, A.; Astrom, K. J., Fast swiyches in relay feedback systems, Automatica, 35, 539-552 (1999) · Zbl 0934.93033 |
| [7] | Malmborg, J.; Bernhardsson, B., Control and simulation of hybrid systems, Communications in Nonlinear Science and Numerical Simulation, 30, 337-347 (1997) · Zbl 0895.93024 |
| [8] | Plathe, E.; Kjøglum, S., Analysis and genetic proporties of gene regulatory networks with graded response functions, Physica D, 201, 150-176 (2005) · Zbl 1078.34015 |
| [9] | Utkin, V. I., Sliding Modes and Their Application in Variable Structure Systems (1978), MIR Publisher: MIR Publisher Moskow · Zbl 0398.93003 |
| [10] | Utkin, V. I., Sliding Mode in Control and Optimization (1992), Springer: Springer Berlin · Zbl 0748.93044 |
| [11] | Casey, R.; de Jong, H.; Gouze, J. L., Piecewise-linear models of genetics regulatory networks: equilibria and their stability, Journal of Mathematical Biology, 52, 27-56 (2006) · Zbl 1091.92030 |
| [12] | de Jong, H.; Gouze, J. L.; Hernandez, C.; Page, M.; Sari, T.; Geiselmann, J., Qualitative simulation of genetic regulatory networks using piecewise-linear models, Bulletin of Mathematical Biology, 66, 301-340 (2004) · Zbl 1334.92282 |
| [13] | Heemels, W. P.M. H.; Schumacher, J. M.; Weiland, S., Linear complementarity systems, SIAM Journal on Applied Mathematics, 60, 4, 1234-1269 (2000) · Zbl 0954.34007 |
| [14] | di Bernardo, M.; Kowalczyk, P.; Nordmark, A., Bifurcations of dynamical systems with sliding: derivation of normal-form mappings, Physica D, 170, 175-205 (2002) · Zbl 1008.37029 |
| [15] | Kuznetsov, Y. A.; Rinaldi, S.; Gragnani, A., One-parameter bifurcations in planar filippov systems, International Journal of Bifurcation and Chaos, 13, 2157-2188 (2003) · Zbl 1079.34029 |
| [16] | Kowalczyk, P.; di Bernardo, M., Two-parameter degenerate sliding bifurcations in filippov systems, Physica D, 204, 204-229 (2005) · Zbl 1087.34019 |
| [17] | Leine, R. I.; Nijmeijer, H., (Dynamics and Bifurcations in Non-Smooth Mechanical Systems. Dynamics and Bifurcations in Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, vol. 18 (2004), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1068.70003 |
| [18] | Leine, R. I.; van Campen, D. H.; van de Vrande, B. L., Bifurcations in nonlinear discontinuous systems, Nonlinear Dynamics, 23, 105-164 (2000) · Zbl 0980.70018 |
| [19] | Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0538.34007 |
| [20] | Filippov, A. F., (Differential Equations with Discontinuous Right-Hand Sides. Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications (1988), Kluwer Academic: Kluwer Academic Dordrecht) · Zbl 0664.34001 |
| [21] | Acary, V.; Brogliato, B., (Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics, Lecture Notes in Applied and Computational Mechanics (2008), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1173.74001 |
| [22] | di Bernardo, M.; Budd, C. J.; Champneys, A. R.; Kowalczyk, P., (Piecewise-smooth Dynamical Systems. Theory and Applications. Piecewise-smooth Dynamical Systems. Theory and Applications, Applied Mathematical Sciences, vol. 163 (2008), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1146.37003 |
| [23] | Llibre, J.; da Silva, P. R.; Teixeira, M. A., Regularization of discontinuous vector fields via singular perturbation, Journal of Dynamics and Differential Equations, 19, 309-331 (2009) · Zbl 1127.34003 |
| [24] | R.I. Leine, Bifurcations in discontinuous mechanical systems of filippov’s type, Ph.D. Thesis, Techn. Univ. Eindhoven, The Netherlands, 2000.; R.I. Leine, Bifurcations in discontinuous mechanical systems of filippov’s type, Ph.D. Thesis, Techn. Univ. Eindhoven, The Netherlands, 2000. |
| [25] | Gear, C. W.; Østerby, O., Solving ordinary differential equations with discontinuities, ACM Transactions on Mathematical Software, 10, 23-24 (1984) · Zbl 0553.65050 |
| [26] | A. Hindmarsh, GEAR: Ordinary differential solver, Tech. Rep. UCID-30001, Revision 3, Lawrence Livermore National Laboratories, Livermore California, 1974.; A. Hindmarsh, GEAR: Ordinary differential solver, Tech. Rep. UCID-30001, Revision 3, Lawrence Livermore National Laboratories, Livermore California, 1974. |
| [27] | Stewart, D. E., Rigid-body dynamics with friction and impact, SIAM Review, 42, 3-39 (2000) · Zbl 0962.70010 |
| [28] | Mannshardt, R., One-step methods of any order for ordinary differential equations with discontinuous right-hand sides, Numerische Mathematik, 31, 131-152 (1978) · Zbl 0373.65037 |
| [29] | Holzhueter, T., Simulation of relay control systems using MATLAB/SIMULINK, Control Engineering Practice, 6, 1089-1096 (1998) |
| [30] | Shampine, L. F.; Thompson, S., Event location for ordinary differential equations, Computer and Mathematics with Applications, 39, 43-54 (2000) · Zbl 0956.65055 |
| [31] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0638.65058 |
| [32] | Esposito, J. M.; Kuman, V., An asynchronous integration and event detection algorithm for simulating Multi-Agent hybrid systems, ACM Transactions on Modeling and Computer Simulation, 14, 4, 363-388 (2004) · Zbl 1390.93416 |
| [33] | Esposito, J. M.; Kuman, V., A state event detection algorithm for numerically simulating hybrid systems with model singularities, ACM Transactions on Modeling and Computer Simulation, 17, 1, 1-22 (2007) · Zbl 1281.93066 |
| [34] | Lambert, J. D., Numerical Methods for Ordinary Differential Systems: The Initial Value Problem (1991), John Wiley: John Wiley London · Zbl 0745.65049 |
| [35] | Calvo, M.; Montijano, J. I.; Randez, L., On the solution of discontinuous IVPs by adaptive Runge-Kutta codes, Numerical Algorithms, 33, 163-182 (2003) · Zbl 1030.65082 |
| [36] | Calvo, M.; Montijano, J. I.; Randez, L., The numerical solution of discontinuous IVPs by Runge-Kutta codes: a review, Boletín de la Sociedad Espanõla Mathemática Aplicada, 44, 33-53 (2008) · Zbl 1242.65139 |
| [37] | Enright, W. H.; Jackson, K. R.; Nørsett, S. P.; Thomsen, P. G., Effective solution of discontinuous IVPs using Runge-Kutta formula pair with interpolants, Applied Mathematics and Computation, 27, 313-335 (1988) · Zbl 0651.65058 |
| [38] | Eich-Soellner, E.; Fuhrer, C., Numerical Methods in Multibody Dynamics (1998), B.G. Teubner: B.G. Teubner Stuttgart, Germany · Zbl 0899.70001 |
| [39] | Guglielmi, N.; Hairer, E., Computing breaking points in implicit delay differential equations, Advances in Computational Mathematics, 29, 229-247 (2008) · Zbl 1160.65041 |
| [40] | G. Grabner, R. Kittinger, A. Kecskemethy, An integration Runge-Kutta and polynomial root finding method for reliable event-driven multibody simulation, in: Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, IFAC, Seville 3-5, 2003.; G. Grabner, R. Kittinger, A. Kecskemethy, An integration Runge-Kutta and polynomial root finding method for reliable event-driven multibody simulation, in: Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, IFAC, Seville 3-5, 2003. |
| [41] | Hairer, E.; Lubich, C.; Wanner, G., Solving Ordinary Differential Equations II: Stiff Problems (2010), Springer-Verlag: Springer-Verlag Berlin |
| [42] | Bellen, A.; Zennaro, M., Numerical Methods for Delay Differential Equations (2003), Clarendon Press: Clarendon Press Oxford · Zbl 0749.65042 |
| [43] | Berardi, M.; Lopez, L., On the continuous extension of Adams-Bashforth methods and the event location in discontinuous ODEs, Applied Mathematics Letters, 25, 6, 995-999 (2012) · Zbl 1242.65128 |
| [44] | Carver, M. B., Efficient implementation over discontinuities in ordinary differential equation simulations, Mathematics and Computers in Simulation, 20, 190-196 (1978) · Zbl 0398.65044 |
| [45] | L. Fridman, Chattering in sliding mode systems and singular perturbation, in: Proc. Int. Symp. on Nonlinear Control Systems, 1995, pp. 725-730.; L. Fridman, Chattering in sliding mode systems and singular perturbation, in: Proc. Int. Symp. on Nonlinear Control Systems, 1995, pp. 725-730. |
| [46] | Nordmark, A. B.; Piiroinen, P. T., Simulation and stability analysis of impacting systems with complete chattering, Nonlinearity, 58, 1, 85-106 (2009) · Zbl 1183.70038 |
| [47] | Piiroinen, P. T.; Kuznetsov, Y. A., An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Transactions on Mathematical Software, 34, 13, 1-24 (2008) · Zbl 1190.65109 |
| [48] | Dieci, L.; Lopez, L., Sliding motion in Filippov differential systems: theoretical results and a computational approach, SIAM Journal on Numerical Analysis, 47, 2023-2051 (2009) · Zbl 1197.34009 |
| [49] | Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1094.65125 |
| [50] | Ascher, U. M.; Petzold, L. R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Society for Industrial and Applied Mathematics (1998) · Zbl 0908.65055 |
| [51] | Dieci, L.; Lopez, L., Sliding motion on discontinuity surfaces of high co-dimension. A construction for selecting a Filippov vector field, Numerische Mathematik, 117, 779-811 (2011) · Zbl 1219.34015 |
| [52] | Esposito, J. M.; Kuman, V.; Pappas, G. J., Accurate event detection for simulating hybrid systems, (Di Benedetto, M. D.; Sangiovanni-Vincitelli, A., HSCC 2001. HSCC 2001, LNCS, vol. 2034 (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg), 204-217 · Zbl 0991.93525 |
| [53] | M. Najaf, A. Azil, R. Nikoukhah, Implementation of continuous-time dynamics in scicos, in: Vlatka Hlupic Alexander Verbraeck Ed., Proceedings 15th European Simulation Symposium, SCS European Council / SCS Europe BVBA, 2003.; M. Najaf, A. Azil, R. Nikoukhah, Implementation of continuous-time dynamics in scicos, in: Vlatka Hlupic Alexander Verbraeck Ed., Proceedings 15th European Simulation Symposium, SCS European Council / SCS Europe BVBA, 2003. |
| [54] | Dieci, L.; Lopez, L., Numerical solution of discontinuous differential systems: approaching the discontinuity from one side, Applied Numerical Mathematics (2011) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
