695 research outputs found
An asymptotic induced numerical method for the convection-diffusion-reaction equation
A parallel algorithm for the efficient solution of a time dependent reaction convection diffusion equation with small parameter on the diffusion term is presented. The method is based on a domain decomposition that is dictated by singular perturbation analysis. The analysis is used to determine regions where certain reduced equations may be solved in place of the full equation. Parallelism is evident at two levels. Domain decomposition provides parallelism at the highest level, and within each domain there is ample opportunity to exploit parallelism. Run time results demonstrate the viability of the method
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
Cumulative reports and publications through December 31, 1988
This document contains a complete list of ICASE Reports. Since ICASE Reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available
Parallel-in-Time Solver for the All-at-Once Runge--Kutta Discretization
In this article, we derive fast and robust parallel-in-time preconditioned
iterative methods for the all-at-once linear systems arising upon
discretization of time-dependent PDEs. The discretization we employ is based on
a Runge--Kutta method in time, for which the development of parallel solvers is
an emerging research area in the literature of numerical methods for
time-dependent PDEs. By making use of classical theory of block matrices, one
is able to derive a preconditioner for the systems considered. The block
structure of the preconditioner allows for parallelism in the time variable, as
long as one is able to provide an optimal solver for the system of the stages
of the method. We thus propose a preconditioner for the latter system based on
a singular value decomposition (SVD) of the (real) Runge--Kutta matrix
. Supposing is invertible,
we prove that the spectrum of the system for the stages preconditioned by our
SVD-based preconditioner is contained within the right-half of the unit circle,
under suitable assumptions on the matrix (the assumptions are well
posed due to the polar decomposition of ). We show the
numerical efficiency of our SVD-based preconditioner by solving the system of
the stages arising from the discretization of the heat equation and the Stokes
equations, with sequential time-stepping. Finally, we provide numerical results
of the all-at-once approach for both problems, showing the speed-up achieved on
a parallel architecture
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review
A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances
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