Sparsing in Real Time Simulation

Modelling of mechatronical systems often leads to large DAEs with stiff components. In real time simulation neither implicit nor explicit methods can cope with such systems in an efficient way: explicit methods have to employ too small steps and implicit methods have to solve too large systems of equations. A solution of this general problem is to use a method that allows manipulations of the Jacobian by computing only those parts that are necessary for the stability of the method. Specifically, manipulation by sparsing aims at zeroing out certain elements of the Jacobian leading t a structure that can be exploited using sparse matrix techniques. The elements to be neglected are chosen by an a priori analysis phase that can be accomplished before the real-time simulaton starts. In this article a sparsing criterion for the linearly implicit Euler method is derived that is based on block diagnonalization and matrix perturbation theory

A Half-Explicit Extrapolation Method for Differential-Algebraic Systems of Indix 3

The present paper is concerned with the numerical solution of differential-algebraic index 3 problems. We study an extrapolation method based on a variant of Euler's rule. This method only treats the algebraic variables implicitly, hence has computational advantages compared to fully implicit schemes. A numerical example in line with the theoretical results is include

Achieving Very High Order for Implicit Explicit Time Stepping: Extrapolation Methods

In this paper we construct extrapolated implicit-explicit time stepping methods that allow to efficiently solve problems with both stiff and non-stiff components. The proposed methods can provide very high order discretizations of ODEs, index-1 DAEs, and PDEs in the method of lines framework. These methods are simple to construct, easy to implement and parallelize. We establish the existence of perturbed asymptotic expansions of global errors, explain the convergence orders of these methods, and explore their linear stability properties. Numerical results with stiff ODEs, DAEs, and PDEs illustrate the theoretical findings and the potential of these methods to solve multiphysics multiscale problems

Aproximación de ecuaciones diferenciales mediante una nueva técnica variacional y aplicaciones

[SPA] En esta Tesis presentamos el estudio teórico y numérico de sistemas de ecuaciones diferenciales basado en el análisis de un funcional asociado de forma natural al problema original. Probamos que cuando se utiliza métodos del descenso para minimizar dicho funcional, el algoritmo decrece el error hasta obtener la convergencia dada la no existencia de mínimos locales diferentes a la solución original. En cierto sentido el algoritmo puede considerarse un método tipo Newton globalmente convergente al estar basado en una linearización del problema. Se han estudiado la aproximación de ecuaciones diferenciales rígidas, de ecuaciones rígidas con retardo, de ecuaciones algebraico‐diferenciales y de problemas hamiltonianos. Esperamos que esta nueva técnica variacional pueda usarse en otro tipo de problemas diferenciales. [ENG] This thesis is devoted to the study and approximation of systems of differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. We prove that in seeking to minimize the error by using standard descent schemes, the procedure can never get stuck in local minima, but will always and steadily decrease the error until getting to the original solution. One main step in the procedure relies on a very particular linearization of the problem, in some sense it is like a globally convergent Newton type method. We concentrate on the approximation of stiff systems of ODEs, DDEs, DAEs and Hamiltonian systems. In all these problems we need to use implicit schemes. We believe that this approach can be used in a systematic way to examine other situations and other types of equations.Universidad Politécnica de Cartagen