×

A general framework of low regularity integrators. (English) Zbl 1486.65127

This paper discusses a low regularity integrators method which allows to numerically approximate the solutions of the evolution equation \(u_t-\mathcal{L}u=f(u,\overline u)\) in \(\mathbb{R}\times \Omega\), where \(\Omega\subset \mathbb{R}^d\) and \(\mathcal{L}\) is a linear operator defined on a complex Hilbert space \(X\) such that (i) \(\mathcal{L}\) comutes with its conjugate; (ii) \(\mathcal{L}\) generates a strongly continuous semigroup of contractions; (iii) \(-\mathcal{L}+\overline{\mathcal{L}}\) generates a group of unitary operations. These assumptions allow the authors to deal in a unified framework with parabolic, dispersive, as well as mixed equations.
The proposed approximating scheme relies on introducing filtered oscillations in order to treat the dominant oscillations, triggered by the operator \(-\mathcal{L}+\overline{\mathcal{L}}\), while only approximating the lower order parts by a stabilized Taylor series expansion. This general approach is applied to the time discretization of various concrete PDEs, such as the nonlinear heat equation, the nonlinear Schrödinger equation, the complex Ginzburg-Landau equation, the half-wave and Klein-Gordon equations, with suitable boundary conditions.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg, 2011. · Zbl 1227.35004
[2] C. Besse, B. Bidégaray, and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40. · Zbl 1026.65073
[3] Y. Bruned and K. Schratz, Resonance Based Schemes for Dispersive Equations via Decorated Trees, preprint, https://arxiv.org/abs/2005.01649 (2021).
[4] E. Celledoni, D. Cohen, and B. Owren, Symmetric exponential integrators with an application to the cubic Schrödinger equation, Found. Comput. Math., 8 (2008), pp. 303-317. · Zbl 1147.65102
[5] E. Faou, Geometric Numerical Integration and Schrödinger Equations, European Mathematical Society Publishing House, Zürich, 2012. · Zbl 1239.65078
[6] E. Hairer, S. P. Nø rsett, and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Springer, Berlin 1993. · Zbl 0789.65048
[7] E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer, Berlin 2006. · Zbl 1094.65125
[8] M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), pp. 209-286. · Zbl 1242.65109
[9] M. Hochbruck, J. Leipold, and A. Ostermann, On the convergence of Lawson methods for semilinear stiff problems, Numer. Math., 145 (2020), pp. 553-580. · Zbl 1453.65269
[10] H. Holden, K. H. Karlsen, K.-A. Lie, and N. H. Risebro, Splitting for Partial Differential Equations with Rough Solutions, European Mathematical Society Publishing House, Zürich, 2010. · Zbl 1191.35005
[11] M. Hofmanová and K. Schratz, An oscillatory integrator for the KdV equation, Numer. Math., 136 (2017), pp. 1117-1137. · Zbl 1454.65034
[12] M. Knöller, A. Ostermann, and K. Schratz, A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data, SIAM J. Numer. Anal., 57 (2019), pp. 1967-1986. · Zbl 1422.65222
[13] B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monogr. Appl. Comput. Math. 14. Cambridge University Press, Cambridge, 2004. · Zbl 1069.65139
[14] J. D. Lawson, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal., 4 (1967), pp. 372-380. · Zbl 0223.65030
[15] D. Li, On Kato-Ponce and fractional Leibniz, Rev. Mat. Iberoam., 35 (2019), pp. 23-100. · Zbl 1412.35261
[16] C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), pp. 2141-2153. · Zbl 1198.65186
[17] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), pp. 341-434. · Zbl 1105.65341
[18] G. Métivier, Para-differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems, CRM Series 5. Edizioni della Normale, Pisa, 2008. · Zbl 1156.35002
[19] A. Ostermann and K. Schratz, Low regularity exponential-type integrators for semilinear Schrödinger equations, Found. Comput. Math., 18 (2018), pp. 731-755. · Zbl 1402.65098
[20] A. Ostermann, F. Rousset, and K. Schratz, Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity, Found. Comput. Math., 21 (2021), pp. 725-765. · Zbl 1486.65208
[21] A. Ostermann, F. Rousset, and K. Schratz, Fourier Integrator for Periodic NLS: Low Regularity Estimates Via Discrete Bourgain Spaces, J. Eur. Math. Soc., to appear. · Zbl 1486.65208
[22] A. Ostermann and C. Su, Two exponential-type integrators for the “good” Boussinesq equation, Numer. Math., 143 (2019), pp. 683-712. · Zbl 1428.35425
[23] J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, 1994. · Zbl 0816.65042
[24] K. Schratz, Y. Wang, and X. Zhao, Low-regularity integrators for nonlinear Dirac equations, Math. Comput., 90 (2021), pp. 189-214. · Zbl 1450.35231
[25] Y. Wu and X. Zhao, Optimal Convergence of a Second Order Low-Regularity Integrator for the KdV Equation, preprint, https://arxiv.org/abs/1910.07367 (2019).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.