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Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs. (English) Zbl 1326.65085
Summary: Rosenbrock-Wanner methods usually have order reduction if they are applied to stiff ordinary differential or differential algebraic equations. Therefore, in several papers further order conditions are derived to reduce this effect. In [the author, J. Comput. Appl. Math. 262, 105–114 (2014; Zbl 1302.65179)] the example of A. Prothero and A. Robinson [Math. Comput. 28, 45–162 (1974; Zbl 0309.65034)] is analysed to find further order conditions. In this paper we consider traditional ROW methods such as ROS3P [J. Lang and J. Verwer, BIT 41, No. 4, 731–738 (2001; Zbl 0996.65099)], ROS3Pw [J. Rang and L. Angermann, BIT 45, No. 4, 761–787 (2005; Zbl 1093.65097)], ROS3PL [J. Lang and D. Teleaga, “Towards a fully space-time adaptive FEM for magnetoquasistatics”, IEEE Trans. Magn. 44, No. 6, 1238–1241 (2008; doi:10.1109/TMAG.2007.914837)], and RODASP [G. Steinebach, “Order-reduction of ROW-methods for DAEs and method of lines applications”, Preprint Nr. 1741. Technische Universität Darmstadt (1995)] and improve these methods such that these further order conditions are satisfied. Numerical examples show the advantages of the new methods.

MSC:
65L04 Numerical methods for stiff equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
65L80 Numerical methods for differential-algebraic equations
Software:
MooNMD; RODAS; ROS3P; UMFPACK
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References:
[1] Hairer, E.; Wanner, G., (Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14, (1996), Springer-Verlag Berlin) · Zbl 0859.65067
[2] Strehmel, K.; Weiner, R., (Linear-Implizite Runge-Kutta-Methoden und ihre Anwendung, Teubner-Texte zur Mathematik, vol. 127, (1992), Teubner Stuttgart) · Zbl 0759.65047
[3] Ostermann, A.; Roche, M., Runge-Kutta methods for partial differential equations and fractional orders of convergence, Math. Comp., 59, 200, 403-420, (1992) · Zbl 0769.65068
[4] Ostermann, A.; Roche, M., Rosenbrock methods for partial differential equations and fractional orders of convergence, SIAM J. Numer. Anal., 30, 4, 1084-1098, (1993) · Zbl 0780.65056
[5] Lang, J.; Verwer, J., ROS3P—an accurate third-order rosenbrock solver designed for parabolic problems, BIT, 41, 4, 730-737, (2001) · Zbl 0996.65099
[6] Rang, J.; Angermann, L., New rosenbrock methods for partial differential algebraic equations of index 1, BIT, 45, 4, 761-787, (2005) · Zbl 1093.65097
[7] Rang, J.; Angermann, L., New rosenbrock methods of order 3 for PDAEs of index 2, Adv. Differ. Equ. Control. Process., 1, 2, 193-217, (2008) · Zbl 1162.65386
[8] John, V.; Matthies, G.; Rang, J., A comparison of time-discretization/linearization approaches for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 195, 5995-6010, (2006) · Zbl 1124.76041
[9] John, V.; Rang, J., Adaptive time step control for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 199, 514-524, (2010) · Zbl 1227.76048
[10] Scholz, S., Order barriers for the B-convergence of ROW methods, Computing, 41, 3, 219-235, (1989) · Zbl 0662.65070
[11] G. Steinebach, Order-reduction of ROW-methods for DAEs and method of lines applications, Preprint 1741, Technische Universität Darmstadt, Darmstadt, 1995.
[12] Prothero, A.; Robinson, A., On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comp., 28, 145-162, (1974) · Zbl 0309.65034
[13] Rang, J., An analysis of the prothero-Robinson example for constructing new DIRK and ROW methods, J. Comput. Appl. Math., 262, 105-114, (2014) · Zbl 1302.65179
[14] Lubich, C.; Ostermann, A., Linearly implicit time discretization of non-linear parabolic equations, IMA J. Numer. Anal., 15, 4, 555-583, (1995) · Zbl 0834.65092
[15] Rang, J., A new stiffly accurate rosenbrock-wanner method for solving the incompressible Navier-Stokes equations, (Ansorge, R.; Bijl, H.; Meister, A.; Sonar, T., Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws, Vol. 120, (2012), Springer Verlag Heidelberg, Berlin), 301-315 · Zbl 1382.65286
[16] J. Rang, The Prothero and Robinson example: Convergence studies for Runge-Kutta and Rosenbrock-Wanner methods, Informatik-Bericht 2014-08, TU Braunschweig, Braunschweig (2014).
[17] Lang, J.; Teleaga, D., Towards a fully space-time adaptive FEM for magnetoquasistatics, IEEE Trans. Magn., 44, 1238-1241, (2008)
[18] Steihaug, T.; Wolfbrandt, A., An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations, Math. Comp., 33, 146, 521-534, (1979) · Zbl 0451.65055
[19] Gustafsson, K.; Lundh, M.; Söderlind, G., A PI stepsize control for the numerical solution of ordinary differential equations, BIT, 28, 2, 270-287, (1988) · Zbl 0645.65039
[20] Lang, J., (Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, vol. 16, (2001), Springer-Verlag Berlin)
[21] Sieber, J., Konvergenzanalyse und numerische tests für die prothero-Robinson-gleichung, (2014), TU Darmstadt, (Master thesis)
[22] J. Verwer, Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines, in: D.F. Griffiths, G.A. Watson (Eds.), Numerical Analysis, in: Pitman Research Notes in Mathematics, Boston, 1986, pp. 220-237. · Zbl 0642.65066
[23] Whitham, G. B., Linear and nonlinear waves, (1974), Wiley-Interscience New York · Zbl 0373.76001
[24] John, V.; Matthies, G., Moonmd—a program package based on mapped finite element methods, Comput. Vis. Sci., 6, 163-170, (2004) · Zbl 1061.65124
[25] Davis, T. A., A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 2, 165-195, (2004) · Zbl 1072.65036
[26] Davis, T. A., Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 2, 166-199, (2004) · Zbl 1072.65037
[27] Schäfer, M.; Turek, S., The benchmark problem flow around a cylinder, (Hirschel, E., Flow Simulation with High-Performance Computers II, Notes on Numerical Fluid Mechanics, vol. 52, (1996), Vieweg), 547-566 · Zbl 0874.76070
[28] John, V., Reference values for drag and lift of a two-dimensional time dependent flow around a cylinder, Internat. J. Numer. Methods Fluids, 44, 777-788, (2004) · Zbl 1085.76510
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