Trigonometric integrators for quasilinear wave equations

Gauckler, Ludwig; Lu, Jianfeng; Marzuola, Jeremy; Rousset, Frédéric; Schratz, Katharina

doi:10.1090/mcom/3339

Trigonometric integrators for quasilinear wave equations
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by Ludwig Gauckler, Jianfeng Lu, Jeremy L. Marzuola, Frédéric Rousset and Katharina Schratz HTML | PDF

Math. Comp. 88 (2019), 717-749 Request permission

Abstract:

Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semidiscretization in time with these integrators is shown for a sufficiently regular exact solution. The time integrators are also combined with a Fourier spectral method into a fully discrete scheme, for which error bounds are provided without requiring any CFL-type coupling of the discretization parameters. The proofs of the error bounds are based on energy techniques and on the semiclassical Gårding inequality.

References

Weizhu Bao and Xuanchun Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math. 120 (2012), no. 2, 189–229. MR 2874965, DOI 10.1007/s00211-011-0411-2
B. Cano, Conservation of invariants by symmetric multistep cosine methods for second-order partial differential equations, BIT 53 (2013), no. 1, 29–56. MR 3029294, DOI 10.1007/s10543-012-0393-1
B. Cano and M. J. Moreta, Multistep cosine methods for second-order partial differential systems, IMA J. Numer. Anal. 30 (2010), no. 2, 431–461. MR 2608467, DOI 10.1093/imanum/drn043
Martina Chirilus-Bruckner, Wolf-Patrick Düll, and Guido Schneider, NLS approximation of time oscillatory long waves for equations with quasilinear quadratic terms, Math. Nachr. 288 (2015), no. 2-3, 158–166. MR 3310504, DOI 10.1002/mana.201200325
Christopher Chong and Guido Schneider, Numerical evidence for the validity of the NLS approximation in systems with a quasilinear quadratic nonlinearity, ZAMM Z. Angew. Math. Mech. 93 (2013), no. 9, 688–696. MR 3105032, DOI 10.1002/zamm.201200068
David Cohen, Ernst Hairer, and Christian Lubich, Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations, Numer. Math. 110 (2008), no. 2, 113–143. MR 2425152, DOI 10.1007/s00211-008-0163-9
Xuanchun Dong, Stability and convergence of trigonometric integrator pseudospectral discretization for $N$-coupled nonlinear Klein-Gordon equations, Appl. Math. Comput. 232 (2014), 752–765. MR 3181310, DOI 10.1016/j.amc.2014.01.144
Willy Dörfler, Hannes Gerner, and Roland Schnaubelt, Local well-posedness of a quasilinear wave equation, Appl. Anal. 95 (2016), no. 9, 2110–2123. MR 3515098, DOI 10.1080/00036811.2015.1089236
Wolf-Patrick Düll, Justification of the nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation, Comm. Math. Phys. 355 (2017), no. 3, 1189–1207. MR 3687215, DOI 10.1007/s00220-017-2966-y
B. García-Archilla, J. M. Sanz-Serna, and R. D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput. 20 (1999), no. 3, 930–963. MR 1648882, DOI 10.1137/S1064827596313851
Ludwig Gauckler, Error analysis of trigonometric integrators for semilinear wave equations, SIAM J. Numer. Anal. 53 (2015), no. 2, 1082–1106. MR 3335499, DOI 10.1137/140977217
L. Gauckler, J. Lu, J. L. Marzuola, F. Rousset, K. Schratz, Trigonometric integrators for quasilinear wave equations, previous version of the present paper, arXiv:1702.02981v1, 2017.
Ludwig Gauckler and Daniel Weiss, Metastable energy strata in numerical discretizations of weakly nonlinear wave equations, Discrete Contin. Dyn. Syst. 37 (2017), no. 7, 3721–3747. MR 3639438, DOI 10.3934/dcds.2017158
Cesáreo González and Mechthild Thalhammer, Higher-order exponential integrators for quasi-linear parabolic problems. Part I: Stability, SIAM J. Numer. Anal. 53 (2015), no. 2, 701–719. MR 3317382, DOI 10.1137/140961845
Cesáreo González and Mechthild Thalhammer, Higher-order exponential integrators for quasi-linear parabolic problems. Part II: Convergence, SIAM J. Numer. Anal. 54 (2016), no. 5, 2868–2888. MR 3546979, DOI 10.1137/15M103384
Volker Grimm and Marlis Hochbruck, Error analysis of exponential integrators for oscillatory second-order differential equations, J. Phys. A 39 (2006), no. 19, 5495–5507. MR 2220772, DOI 10.1088/0305-4470/39/19/S10
Mark D. Groves and Guido Schneider, Modulating pulse solutions for quasilinear wave equations, J. Differential Equations 219 (2005), no. 1, 221–258. MR 2181036, DOI 10.1016/j.jde.2005.01.014
Ernst Hairer and Christian Lubich, Long-time energy conservation of numerical methods for oscillatory differential equations, SIAM J. Numer. Anal. 38 (2000), no. 2, 414–441. MR 1770056, DOI 10.1137/S0036142999353594
Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration, 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, 2006. Structure-preserving algorithms for ordinary differential equations. MR 2221614
Daniel Han-Kwan and Frédéric Rousset, Quasineutral limit for Vlasov-Poisson with Penrose stable data, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 6, 1445–1495 (English, with English and French summaries). MR 3592362, DOI 10.24033/asens.2313
Marlis Hochbruck and Tomislav Pažur, Error analysis of implicit Euler methods for quasilinear hyperbolic evolution equations, Numer. Math. 135 (2017), no. 2, 547–569. MR 3599569, DOI 10.1007/s00211-016-0810-5
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, DOI 10.1007/BF00251584
Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
Balázs Kovács and Christian Lubich, Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type, Numer. Math. 138 (2018), no. 2, 365–388. MR 3748307, DOI 10.1007/s00211-017-0909-3
P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
Jianfeng Lu and Jeremy L. Marzuola, Strang splitting methods for a quasilinear Schrödinger equation: convergence, instability, and dynamics, Commun. Math. Sci. 13 (2015), no. 5, 1051–1074. MR 3344418, DOI 10.4310/CMS.2015.v13.n5.a1
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
Christopher D. Sogge, Lectures on nonlinear wave equations, Monographs in Analysis, II, International Press, Boston, MA, 1995. MR 1715192
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
Michael E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. MR 1121019, DOI 10.1007/978-1-4612-0431-2
Michael E. Taylor, Partial differential equations III. Nonlinear equations, 2nd ed., Applied Mathematical Sciences, vol. 117, Springer, New York, 2011. MR 2744149, DOI 10.1007/978-1-4419-7049-7