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New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index. (English) Zbl 1093.65097
New Rosenbrock methods for ordinary differential equations, differential-algebraic equations; partial differential equations (PDEs) and partial differential-algebraic equations (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 of the five new methods with five other Rosenbrock solvers shows the advantages of the new methods. They are applied to several differential equations such as a PDE, an index-1 PDAE and the Navier-Stokes equations with different right-hand sides. The results are presented in pleasant figures. The numerically observed temporal order of convergence are also given for the different examples.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65L80 Numerical methods for differential-algebraic equations 34A09 Implicit ordinary differential equations, differential-algebraic equations 35K55 Nonlinear parabolic equations 35R10 Partial functional-differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65L20 Stability and convergence of numerical methods for ordinary differential equations 35Q30 Navier-Stokes equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
##### Software:
DASSL; MooNMD; ROWMAP; RODAS; VODE; ROS3P
Full Text:
##### References:
 [1] 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. · Zbl 0677.65075 [2] 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. · Zbl 0844.65058 [3] F. Cameron, A class of low order DIRK methods for a class of DAEs, Appl. Numer. Math., 31 (1999), pp. 1–16. · Zbl 0945.65094 [4] A. J. Chorin, Numerical solution for the Navier–Stokes equations, Math. Comput., 22 (1968), pp. 745–762. · Zbl 0198.50103 [5] E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier–Stokes-Problems, PhD thesis, Technische Universität Berlin, 2001. Appeared also as book from Cuvillier Verlag Göttingen. · Zbl 0982.76003 [6] E. Hairer, C. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods. Springer, Berlin, 1989. · Zbl 0683.65050 [7] M. Hochbruck, C. Lubich and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19 (1998), pp. 1552–1574. · Zbl 0912.65058 [8] 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. · Zbl 0859.65067 [9] V. John and G. Matthies, MooNMD – a program package based on mapped finite element methods, Comput. Vis. Sci., 6 (2004), pp. 163–170. · Zbl 1061.65124 [10] 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. · Zbl 1085.76510 [11] C. Lubich and A. Ostermann, Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal., 15 (1995), pp. 555–583. · Zbl 0834.65092 [12] 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. · Zbl 0692.65038 [13] J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, vol. 16, Springer, Berlin, 2001. · Zbl 0963.65102 [14] J. Lang and J. Verwer, ROS3P – an accurate third-order Rosenbrock solver designed for parabolic problems, BIT, 41 (2001), pp. 730–737. · Zbl 0996.65099 [15] A. Ostermann, Über die Wahl geeigneter Approximationen an die Jacobimatrix bei linear-impliziten Runge–Kutta Verfahren, PhD thesis, Universität Innsbruck, 1988. [16] 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. [17] J. Rang and L. Angermann, Perturbation index of linear partial differential algebraic equations, Appl. Numer. Math., 53 (2005), pp. 437–456. · Zbl 1073.35053 [18] M. Roche, Implicit Runge–Kutta methods for differential algebraic equations, SIAM J. Numer. Anal., 26 (1989), pp. 963–975. · Zbl 0674.65040 [19] 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. [20] 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. · Zbl 0874.76070 [21] G. Steinebach, Order-reduction of ROW-methods for DAEs and method of lines applications, Preprint 1741, Technische Universität Darmstadt, Darmstadt, 1995. [22] 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. · Zbl 0451.65055 [23] K. Strehmel and R. Weiner, Linear-implizite Runge–Kutta-Methoden und ihre Anwendung, Teubner-Texte zur Mathematik, vol. 127, Teubner, Stuttgart, 1992. [24] 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. · Zbl 0928.65116 [25] 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. · Zbl 0895.65035
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