Convergence of Runge-Kutta methods for differential-algebraic systems of index 3
type: Item
instance of: scholarly article
title: Convergence of Runge-Kutta methods for differential-algebraic systems of index 3
zbMATH Open document ID: 0832.65078
DOI: 10.1016/0168-9274(95)00013-K
author: Laurent O. Jay
published in: Applied Numerical Mathematics
publication date: 1996-03-05T00:00:00Z
review text: Index 3 differential-algebraic equations (DAEs) are notoriously difficult to solve directly and it is a common practice to resort to index reduction. However, in the modeling of many mechanical systems, this is not an appropriate approach, and it is necessary to retain the index 3 formulation. This paper considers the rate of convergence of solutions obtained by the use of implicit Runge-Kutta methods applied to semi- explicit index-3 DAEs in Hessenberg form, \(y' = f(y,z)\), \(z' = k(y, z, u)\), \(0 = g(y)\), where \(g_y f_z k_u\) is invertible in the vicinity of the exact solution. In particular the paper examines the order enhancement that can be obtained by the application of a projection to the \(z\) component, with a corresponding adjustment to the \(u\) component. Denote by \(B(p)\) (\(p\) is the quadrature order), \(C(q)\) (\(q\) is the stage order) and \(D(r)\) the usual simplifying assumptions, so that these hold with \(p =2s - 2\), \(q = r = s - 1\) in the case of Lobatto IIIC methods and with \(p =2s -1\), \(q = s\), \(r = s - 1\) in the case of Radau IIA methods. After a number of preliminary results preparing the ground, the principal theorem and its corollaries show that under certain assumptions on the initial value problem and on the coefficients of the Runge-Kutta method, the global error in the \(y\) component has order \(\min(p,2q - 2, q + r)\), with a slightly improved estimate if \(k\) is linear in \(u\). Moreover, although the global errors in \(z\) and \(u\) are limited to \(q\) and \(q - 1\) respectively, the use of a projection raises these to the same order as for the \(y\) component. The results are supported by numerical experiments in a variable stepsize setting for both a linear (\(k\) linear in \(u\)) problem and a nonlinear problem, using Lobatto IIIC and Radau IIA methods of various classical orders.
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 770317
zbMATH Keywords: global superconvergence
zbMATH Keywords: index 3 differential-algebraic equations
zbMATH Keywords: index reduction
zbMATH Keywords: convergence
zbMATH Keywords: implicit Runge-Kutta methods
zbMATH Keywords: Hessenberg form
zbMATH Keywords: Lobatto IIIC methods
zbMATH Keywords: Radau IIA methods
zbMATH Keywords: numerical experiments
zbMATH Keywords: variable stepsize
reviewed by: J. C. Butcher
describes a project that uses: RODAS
MaRDI profile type: Publication
cites work: Projected Implicit Runge–Kutta Methods for Differential-Algebraic Equations
cites work: Q3993493
cites work: Backward differentiation approximations of nonlinear differential/algebraic systems
cites work: The Numerical Solution of Higher Iindex Differential/Algebraic Equations by Implicit Methods
cites work: Coefficients for the study of Runge-Kutta integration processes
cites work: Q4725642
cites work: Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations
cites work: Q3550030
cites work: Q4000061
cites work: Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2
cites work: Collocation methods for differential-algebraic equations of index 3
cites work: Q4339211
cites work: Integration of stiff mechanical systems by Runge-Kutta methods
cites work: A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3
cites work: Implicit Runge–Kutta Methods for Differential Algebraic Equations
Wikidata QID: Q127981342
description: scientific article; zbMATH DE number 770317