Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

Gauckler, Ludwig J.; Weiß, Daniel

doi:10.3934/dcds.2017158

Cited by 10 publications

(7 citation statements)

References 21 publications

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“…• filter functions (13), which is the method of Hairer & Lubich and coincides with the new method (15) for c = 0, Different lines correspond to different values of the discretization parameter K = 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , with darker lines for larger values of K.…”

Section: Statement Of Global Error Boundsmentioning

confidence: 91%

Trigonometric integrators for quasilinear wave equations

Gauckler

Marzuola

et al. 2018

Math. Comp.

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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.Mathematics Subject Classification (2010): 65M15, 65P10, 65L70, 65M20.

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Section: Statement Of Global Error Boundsmentioning

confidence: 91%

Trigonometric integrators for quasilinear wave equations

Gauckler

Marzuola

et al. 2018

Math. Comp.

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“…Theorem 1 (See [21].) Under the non-resonance condition (9) and the assumption (16) given in [21], it holds that…”

Section: Full Discretisation 21 Spectral Semi-discretisation In Spacementioning

confidence: 99%

“…Inserting the modulated Fourier expansions ( 14), (16), and ( 17) into (15) yields q(t + h) − 2 cos(hΩ)q(t) + q(t − h)…”

Section: Modulation Equationsmentioning

confidence: 99%

“…In recent decades many numerical methods have been developed and researched for solving wave equations (see, e.g. [3,4,5,10,16,17,18,24,30,33]). As one important aspect of the analysis, long-time conservation properties of wave equations or of some numerical methods applied to wave equations have been well studied and we refer the reader to [9,8,12,13,21].…”

Section: Introductionmentioning

confidence: 99%

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Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

Wang

2019

Adv Comput Math

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As is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semilinear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. Both the results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First a multi-frequency modulated Fourier expansion of the AAVF method is constructed and then two almost-invariants of the modulation system are derived.

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“…These methods were also shown to work well for wave equations in the semilinear case (see, e.g. [1,2,3,4,12]). Recently, a standard form of trigonometric integrators called as ERKN integrators was formulated for solving secondorder highly oscillatory differential equations in [44], which can be of arbitrarily high order.…”

Section: Introductionmentioning

confidence: 99%

Functionally-fitted energy-preserving integrators for Poisson systems

Wang

2018

Journal of Computational Physics

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In this paper, we present an error analysis of one-stage explicit extended Runge-Kutta-Nyström integrators for semilinear wave equations. These equations are analysed by using spatial semidiscretizations with periodic boundary conditions in one space dimension. Optimal second-order convergence is proved without requiring Lipschitz continuous and higher regularity of the exact solution. Moreover, the error analysis is not restricted to the spectral semidiscretization in space.

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Metastable energy strata in numerical discretizations of weakly nonlinear wave equations

Cited by 10 publications

References 21 publications

Trigonometric integrators for quasilinear wave equations

Trigonometric integrators for quasilinear wave equations

Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

Functionally-fitted energy-preserving integrators for Poisson systems

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