Convergence of Runge-Kutta methods for differential-algebraic systems of index 3. (English) Zbl 0832.65078

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.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems


Full Text: DOI


[1] Ascher, U.M; Petzold, L.R, Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. numer. anal., 28, 1097-1120, (1991) · Zbl 0732.65067
[2] Brenan, K.E; Campbell, S.L; Petzold, L.R, Numerical solution of initial-value problems in differential-algebraic equations, (1989), North-Holland New York · Zbl 0699.65057
[3] Brenan, K.E; Engquist, B.E; Brenan, K.E; Engquist, B.E, Backward differentiation approximations of nonlinear differential/algebraic systems, Math. comp., Supplement math. comp., 51, S7-S16, (1988) · Zbl 0699.65059
[4] Brenan, K; Petzold, L.R, The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods, SIAM J. numer. anal., 26, 976-996, (1989) · Zbl 0681.65050
[5] Butcher, J.C, Coefficients for the study of Runge-Kutta integration processes, J. austral. math. soc., 3, 185-201, (1963) · Zbl 0223.65031
[6] Butcher, J.C, The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods, (1987), Wiley Chichester, England · Zbl 0616.65072
[7] Hairer, E; Lubich, C; Roche, M, The numerical solution of differential-algebraic systems by Runge-Kutta methods, () · Zbl 0657.65093
[8] Hairer, E; Nørsett, S.P; Wanner, G, Solving ordinary differential equations I. nonstiff problems, () · Zbl 1185.65115
[9] Hairer, E; Wanner, G, Solving ordinary differential equations II. stiff and differential-algebraic problems, () · Zbl 0729.65051
[10] Jay, L, Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2, Bit, 33, 137-150, (1993) · Zbl 0773.65049
[11] Jay, L, Collocation methods for differential-algebraic equations of index 3, Numer. math., 65, 407-421, (1993) · Zbl 0791.65056
[12] Jay, L, Runge-Kutta type methods for index three differential-algebraic equations with applications to Hamiltonian systems, ()
[13] Lubich, C, Integration of stiff mechanical systems by Runge-Kutta methods, Z. angew. math. phys., 44, 1022-1053, (1993) · Zbl 0784.70002
[14] Ostermann, A, A class of half-explicit Runge-Kutta methods for differential-algebraic systems of index 3, Appl. numer. math., 13, 165-179, (1993) · Zbl 0788.65084
[15] Roche, M, Implicit Runge-Kutta for differential algebraic equations, SIAM J. numer. anal., 26, 963-975, (1989) · Zbl 0674.65040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.