Abstract
In this note new Rosenbrock methods for ODEs, DAEs, PDEs and PDAEs of index 1 are presented. These solvers are of order 3, have 3 or 4 internal stages, and fulfil certain order conditions to obtain a better convergence if inexact Jacobians and approximations of \(\frac{\partial f}{\partial t}\) are used. A comparison with other Rosenbrock solvers shows the advantages of the new methods.
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G. D. Byrne, P. N. Brown and A. C. Hindmarsh, VODE: a variable-coefficient ODE solver, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 1038–1051.
K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical solution of initial-value problems in DAEs, Classics in Applied Mathematics, vol. 14, SIAM, Philadelphia, 1996.
F. Cameron, A class of low order DIRK methods for a class of DAEs, Appl. Numer. Math., 31 (1999), pp. 1–16.
A. J. Chorin, Numerical solution for the Navier–Stokes equations, Math. Comput., 22 (1968), pp. 745–762.
E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier–Stokes-Problems, PhD thesis, Technische Universität Berlin, 2001. Appeared also as book from Cuvillier Verlag Göttingen.
E. Hairer, C. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods. Springer, Berlin, 1989.
M. Hochbruck, C. Lubich and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552–1574.
E. Hairer and G. Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, vol. 14, 2nd edition, Springer, Berlin, 1996.
V. John and G. Matthies, MooNMD — a program package based on mapped finite element methods, Comput. Vis. Sci., 6 (2004), pp. 163–170.
V. John, Reference values for drag and lift of a two-dimensional time dependent flow around a cylinder, Int. J. Numer. Methods Fluids, 44 (2004), pp. 777–788.
C. Lubich and A. Ostermann, Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal., 15 (1995), pp. 555–583.
C. Lubich and M. Roche, Rosenbrock methods for differential-algebraic systems with solution-dependent singular matrix multiplying the derivative. Computing, 43 (1990), pp. 325–342.
J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, vol. 16, Springer, Berlin, 2001.
J. Lang and J. Verwer, ROS3P — an accurate third-order Rosenbrock solver designed for parabolic problems, BIT, 41 (2001), pp. 730–737.
A. Ostermann, Über die Wahl geeigneter Approximationen an die Jacobimatrix bei linear-impliziten Runge–Kutta Verfahren, PhD thesis, Universität Innsbruck, 1988.
J. Rang, Stability estimates and numerical methods for degenerate parabolic differential equations, PhD thesis, Institut für Mathematik, TU Clausthal, 2004. Appeared also as book from Papierflieger Verlag, Clausthal, 2005.
J. Rang and L. Angermann, Perturbation index of linear partial differential algebraic equations, Appl. Numer. Math., 53 (2005), pp. 437–456.
M. Roche, Implicit Runge–Kutta methods for differential algebraic equations, SIAM J. Numer. Anal., 26 (1989), pp. 963–975.
A. Sandu, J. G. Verwer, J. G. Blom, E. J. Spee, C. R. Carmichael and F. A. Potra, Benchmarking stiff ODE solvers for athmospheric chemistry problems II: Rosenbrock solves, Atmos. Environ., 31 (1997), pp. 3459–3472.
M. Schäfer and S. Turek, The benchmark problem “Flow around a cylinder", in E. H. Hirschel, editor, Flow simulation with high-performance computers II, Notes on Numerical Fluid Mechanics, vol. 52, pp. 547–566. Vieweg, 1996.
G. Steinebach, Order-reduction of ROW-methods for DAEs and method of lines applications, Preprint 1741, Technische Universität Darmstadt, Darmstadt, 1995.
T. Steihaug and A. Wolfbrandt, An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations, Math. Comput., 33 (1979), pp. 521–534.
K. Strehmel and R. Weiner, Linear-implizite Runge–Kutta-Methoden und ihre Anwendung, Teubner-Texte zur Mathematik, vol. 127, Teubner, Stuttgart, 1992.
J. Verwer, E. J. Spee, J. G. Blom and W. Hundsdorfer, A second-order Rosenbrock method applied to photochemical dispersion problems, SIAM J. Sci. Comput., 20 (1999), pp. 1456–1480.
R. Weiner, B. Schmitt and H. Podhaisky, ROWMAP – a ROW-code with Krylov techniques for large stiff ODEs, Appl. Numer. Math., 25 (1997), pp. 303–319.
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AMS subject classification (2000)
34A09, 65L80
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Rang, J., Angermann, L. New Rosenbrock W-Methods of Order 3 for Partial Differential Algebraic Equations of Index 1. Bit Numer Math 45, 761–787 (2005). https://doi.org/10.1007/s10543-005-0035-y
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DOI: https://doi.org/10.1007/s10543-005-0035-y