ISBN 978-3-8439-3134-2

978-3-8439-3134-2, Reihe Mathematik

Alexander Rath
Global Error Estimation for Stiff Differential Equations

105 Seiten, Dissertation Technische Universität Darmstadt (2016), Softcover, A5

Zusammenfassung / Abstract

In this work, we present an approach for global error estimation in initial value problems and analyze it in the context of stiff equations. For two instances we prove asymptotic correctness when applied to arbitrary solvers up to order one and three, respectively.

The approach is based on constructing a continuous extension to the discrete approximations. We derive a continuous error function which itself is solution of a transport equation. Solving this secondary problem yields approximations of the true error. We consider linear splines and piecewise cubic Hermite interpolation to obtain the error function and solve a linearization of the error equation using the implicit midpoint rule. This Gaussian scheme features valuable stability properties, which are passed on to the estimates.

Since stiff problems often generate unpleasant effects like order reduction, asymptotic arguments from standard theory of ordinary differential equations (ODEs) should not be trusted blindly.

We analyze the presented methods using the concepts of B-consistency and B-convergence to derive convergence results independent of stiffness. Due to its structure, the implicit midpoint rule shows converge rates of high order when applied to the error equation. If certain derivatives of the exact solution and the right hand side are bounded by moderate constants, the convergence rate of the estimates is restricted mainly by the accuracy of the continuous extension and the order of the primary method. Although these prerequisites restrict the class of stiff systems, many problems of practical relevance are covered.

Numerical results are presented for parabolic and hyperbolic problems in one and two spatial dimensions. The equations are discretized using finite differences and finite elements and the resulting stiff ODEs are solved using Runge-Kutta methods of several orders.