Convergence of Time-Stepping Schemes for Passive and Extended Linear Complementarity Systems

Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established.</p

Convergence of time - stepping schemes for passıie and extended linear complementarity systems

Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established

Complementarity methods in the analysis of piecewise linear dynamical systems

The main object of this thesis is a class of piecewise linear dynamical systems that are related both to system theory and to mathematical programming. The dynamical systems in this class are known as complementarity systems. With regard to these nonlinear and nonsmooth dynamical systems, the research in the thesis concentrates on two themes: well-posedness and approximations. The well-posedness issue, in the sense of existence and uniqueness of solutions, is of considerable importance from a model validation point of view. In the thesis, sufficient conditions are established for the well-posedness of complementarity systems. Furthermore, an investigation is made of the convergence of approximations of these systems with an eye towards simulation

Differential variational inequalities

International audienceThis paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems , and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensiona

An All-At-Once Newton Strategy for Marine Methane Hydrate Reservoir Models

The migration of methane through the gas hydrate stability zone (GHSZ) in the marine subsurface is characterized by highly dynamic reactive transport processes coupled to thermodynamic phase transitions between solid gas hydrates, free methane gas, and dissolved methane in the aqueous phase. The marine subsurface is essentially a water-saturated porous medium where the thermodynamic instability of the hydrate phase can cause free gas pockets to appear and disappear locally, causing the model to degenerate. This poses serious convergence issues for the general-purpose nonlinear solvers (e.g., standard Newton), and often leads to extremely small time-step sizes. The convergence problem is particularly severe when the rate of hydrate phase change is much lower than the rate of gas dissolution. In order to overcome this numerical challenge, we have developed an all-at-once Newton scheme tailored to our gas hydrate model, which can handle rate-based hydrate phase change coupled with equilibrium gas dissolution in a mathematically consistent and robust manner

Time-stepping numerical simulation of switched circuits with the nonsmooth dynamical systems approach

International audienceThe numerical integration of switching circuits is known to be a tough issue when the number of switches is large, or when sliding modes exist. Then, classical analog simulators may behave poorly, or even fail. In this paper, it is shown on two examples that the nonsmooth dynamical systems (NSDS) approach, which is made of: 1) a specific modeling of the piecewise-linear electronic devices (ideal diodes, Zener diodes, transistors); 2) the Moreau's time-stepping scheme; and 3) specific iterative one-step solvers, supersedes simulators of the simulation program with integrated circuit emphasis (SPICE) family and hybrid simulators. An academic example constructed in [Maffezzoni, , IEEE Trans. CADICS, vol 25, no. 11, Nov. 2006], so that the Newton-Raphson scheme does not converge, and the buck converter are used to make extensive comparisons between the NSDS method and other methods of the SPICE family and a hybrid-like method. The NSDS method, implemented in the siconos platform developed at INRIA, proves to be on these two examples much faster and more robust with respect to the model parameter variations