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An Attempt to Avoid Exact Jacobian in the Numerical Solution of Differential-Algebraic Equations by W-Methods

  • Solving differential-algebraic equations (DAEs) efficiently is an ongoing topic in applied mathematics. Applications are given with respect to many fields of practical interest, such as multiphysics problems or network simulations. Due to the stiffness properties of DAEs, linearly implicit Runge-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes are an appropriate choice for effecitive numerical time-integration. Compared to fully implicit schemes, they are easy to implement and avoid having to solve non-linear equations by including Jacobian information in their formulation explicity. But, especially when having to solve large coupled systems, computing the Jacobian is costly and proves to be a considerable drawback. Inspired by the works of Steihaug and Wolfbrandt [4], we introduce concepts to realize linearly-implicit Runge-Kutta methods for DAEs in the form of so-called W-methods. These schemes allow for arbitrary approximations to given Jacobian entries and, thus, for versatile strategies to reduce computational effort significantly when solving semi-explicit DAE problems of index-1. An approach extending Roche’s procedure [3] will be presented that enables to derive order conditions of the resulting methods by an algebraic theory using rooted trees, a strategy originally introduced by Butcher regarding Runge-Kutta schemes [1,2]. Besides, suitable sets of coefficients for implementing embedded schemes and their potential of increasing efficincy when solving DAEs will be demonstrated.

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Document Type:Conference Object
Author:Tim Jax
Parent Title (English):SciCADE 2017, International Conference on Scientific Computation and Differential Equations, University of Bath, UK, September 11-15, 2017. Book of Abstracts
First Page:136
Publication year:2017
Award:John Butcher Prize in Numerical Analysis
Tag:Approximated Jacobian; Rosenbrock-Wanner Methods; Theory of Rooted Trees; W-Methods; differential-algebraic equations
Departments, institutes and facilities:Fachbereich Elektrotechnik, Maschinenbau, Technikjournalismus
Institut für Technik, Ressourcenschonung und Energieeffizienz (TREE)
Dewey Decimal Classification (DDC):5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Entry in this database:2017/09/23