Runge-Kutta methods for linear semi-explicit operator differential-algebraic equations

Altmann, R.; Zimmer, C.

doi:10.1090/mcom/3270

Runge-Kutta methods for linear semi-explicit operator differential-algebraic equations
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by R. Altmann and C. Zimmer;

Math. Comp. 87 (2018), 149-174

DOI: https://doi.org/10.1090/mcom/3270

Published electronically: June 21, 2017

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Abstract:

As a first step towards time-stepping schemes for constrained PDE systems, this paper presents convergence results for the temporal discretization of operator DAEs. We consider linear, semi-explicit systems which include e.g. the Stokes equations or applications with boundary control. To guarantee unique approximations, we restrict the analysis to algebraically stable Runge-Kutta methods for which the stability functions satisfy $R(\infty )=0$. As expected from the theory of DAEs, the convergence properties of the single variables differ and depend strongly on the assumed smoothness of the data.

References

R. Altmann and J. Heiland, Finite element decomposition and minimal extension for flow equations, ESAIM Math. Model. Numer. Anal. 49 (2015), no. 5, 1489–1509. MR 3423233, DOI 10.1051/m2an/2015029
Robert Altmann, Index reduction for operator differential-algebraic equations in elastodynamics, ZAMM Z. Angew. Math. Mech. 93 (2013), no. 9, 648–664. MR 3105029, DOI 10.1002/zamm.201200125
R. Altmann, Regularization and Simulation of Constrained Partial Differential Equations, Ph.D. thesis, Technische Universität Berlin, 2015.
M. Arnold, Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index $2$, BIT 38 (1998), no. 3, 415–438. MR 1652828, DOI 10.1007/BF02510252
Kevin Burrage and J. C. Butcher, Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979), no. 1, 46–57. MR 518683, DOI 10.1137/0716004
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Michele Benzi, Gene H. Golub, and Jörg Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005), 1–137. MR 2168342, DOI 10.1017/S0962492904000212
P. Benner and J. Heiland, Time-dependent Dirichlet conditions in finite element discretizations, Science Open Research (2015).
Adam Bowers and Nigel J. Kalton, An introductory course in functional analysis, Universitext, Springer, New York, 2014. With a foreword by Gilles Godefroy. MR 3289046, DOI 10.1007/978-1-4939-1945-1
Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235, DOI 10.1017/CBO9780511618635
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
Randolph E. Bank, Bruno D. Welfert, and Harry Yserentant, A class of iterative methods for solving saddle point problems, Numer. Math. 56 (1990), no. 7, 645–666. MR 1031439, DOI 10.1007/BF01405194
Etienne Emmrich and Volker Mehrmann, Operator differential-algebraic equations arising in fluid dynamics, Comput. Methods Appl. Math. 13 (2013), no. 4, 443–470. MR 3107356, DOI 10.1515/cmam-2013-0018
E. Emmrich, Analysis von Zeitdiskretisierungen des inkompressiblen Navier-Stokes-Problems, Cuvillier, Göttingen, 2001.
E. Emmrich, Gewöhnliche und Operator-Differentialgleichungen: Eine Integrierte Einführung in Randwertprobleme und Evolutionsgleichnugen für Studierende, Vieweg, Wiesbaden, 2004.
Etienne Emmrich and Mechthild Thalhammer, Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math. 10 (2010), no. 2, 171–190. MR 2594443, DOI 10.1007/s10208-010-9061-5
C. W. Gear, B. Leimkuhler, and G. K. Gupta, Automatic integration of Euler-Lagrange equations with constraints, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 77–90. MR 793945, DOI 10.1016/0377-0427(85)90008-1
Sara Grundel, Lennart Jansen, Nils Hornung, Tanja Clees, Caren Tischendorf, and Peter Benner, Model order reduction of differential algebraic equations arising from the simulation of gas transport networks, Progress in differential-algebraic equations, Differ.-Algebr. Equ. Forum, Springer, Heidelberg, 2014, pp. 183–205. MR 3588250
Eberhard Griepentrog and Roswitha März, Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 88, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1986. With German, French and Russian summaries. MR 881052
Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991. MR 1091716, DOI 10.1017/CBO9780511840371
Ernst Hairer, Christian Lubich, and Michel Roche, The numerical solution of differential-algebraic systems by Runge-Kutta methods, Lecture Notes in Mathematics, vol. 1409, Springer-Verlag, Berlin, 1989. MR 1027594, DOI 10.1007/BFb0093947
M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE constraints, Mathematical Modelling: Theory and Applications, vol. 23, Springer, New York, 2009. MR 2516528
J. C. Heinrich and C. A. Vionnet, The penalty method for the Navier-Stokes equations, Arch. Comput. Methods Engrg. 2 (1995), no. 2, 51–65. MR 1343072, DOI 10.1007/BF02904995
E. Hairer and G. Wanner, Solving ordinary differential equations. II, 2nd ed., Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1996. Stiff and differential-algebraic problems. MR 1439506, DOI 10.1007/978-3-642-05221-7
Lennart Jansen and Caren Tischendorf, A unified (P)DAE modeling approach for flow networks, Progress in differential-algebraic equations, Differ.-Algebr. Equ. Forum, Springer, Heidelberg, 2014, pp. 127–151. MR 3588248
Peter Kunkel and Volker Mehrmann, Differential-algebraic equations, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2006. Analysis and numerical solution. MR 2225970, DOI 10.4171/017
René Lamour, Roswitha März, and Caren Tischendorf, Differential-algebraic equations: a projector based analysis, Differential-Algebraic Equations Forum, Springer, Heidelberg, 2013. MR 3024597, DOI 10.1007/978-3-642-27555-5
Ch. Lubich and A. Ostermann, Runge-Kutta methods for parabolic equations and convolution quadrature, Math. Comp. 60 (1993), no. 201, 105–131. MR 1153166, DOI 10.1090/S0025-5718-1993-1153166-7
Christian Lubich and Alexander Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Math. Comp. 64 (1995), no. 210, 601–627. MR 1284670, DOI 10.1090/S0025-5718-1995-1284670-0
V. Mehrmann, Index Concepts for Differential-Algebraic Equations, Encyclopedia of Applied and Computational Mathematics (T. Chan, W.J. Cook, E. Hairer, J. Hastad, A. Iserles, H.P. Langtangen, C. Le Bris, P.L. Lions, C. Lubich, A.J. Majda, J. McLaughlin, R.M. Nieminen, J. Oden, P. Souganidis, and A. Tveito, eds.), Springer-Verlag, Berlin, 2013.
Solomon G. Michlin, Variationsmethoden der mathematischen Physik, Akademie-Verlag, Berlin, 1962 (German). In deutscher Sprache herausgegeben von Eugen Heyn. MR 141248
L. R. Petzold, Order results for implicit Runge-Kutta methods applied to differential/algebraic systems, SIAM J. Numer. Anal. 23 (1986), no. 4, 837–852. MR 849286, DOI 10.1137/0723054
Tomáš Roubíček, Nonlinear partial differential equations with applications, International Series of Numerical Mathematics, vol. 153, Birkhäuser Verlag, Basel, 2005. MR 2176645
Jie Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J. Numer. Anal. 32 (1995), no. 2, 386–403. MR 1324294, DOI 10.1137/0732016
B. Simeon, Numerische Simulation gekoppelter Systeme von partiellen und differential-algebraischen Gleichungen der Mehrkörperdynamik, VDI Verlag, Düsseldorf, 2000.
Bernd Simeon, Computational flexible multibody dynamics, Differential-Algebraic Equations Forum, Springer, Heidelberg, 2013. A differential-algebraic approach. MR 3086702, DOI 10.1007/978-3-642-35158-7
Luc Tartar, An introduction to Navier-Stokes equation and oceanography, Lecture Notes of the Unione Matematica Italiana, vol. 1, Springer-Verlag, Berlin; UMI, Bologna, 2006. MR 2258988, DOI 10.1007/3-540-36545-1
Luc Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana, vol. 3, Springer, Berlin; UMI, Bologna, 2007. MR 2328004
Roger Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 609732
C. Tischendorf, Solution of index-2 differential algebraic equations and its application in circuit simulation, Ph.D. thesis, Humboldt-Universität zu Berlin, 1996.
C. Tischendorf, Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation. Modeling and Numerical Analysis, Habilitationsschrift, Humboldt-Universität zu Berlin, 2003.
J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR 895589, DOI 10.1017/CBO9781139171755
Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990. Linear monotone operators; Translated from the German by the author and Leo F. Boron. MR 1033497, DOI 10.1007/978-1-4612-0985-0