Trigonometric integrators for quasilinear wave equations

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

doi:10.1090/mcom/3339

Cited by 18 publications

(29 citation statements)

References 31 publications

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“…for −1 ≤ β ≤ 1 and all ξ = hω j with j ∈ K and ω j = 0. Moreover, we assume that c 1 = 1 2 for the ERKN integrator (11). Table 1 satisfy this assumption uniformly for −1 ≤ β ≤ 1 and h > 0.…”

Section: Resultsmentioning

confidence: 99%

“…This subsection studies the stability of the ERKN integrator (11). Proof For (w ⊺ ,ẇ ⊺ ) ⊺ = R(h)(v ⊺ ,v ⊺ ) ⊺ , it has been shown in [9] that…”

Section: Stabilitymentioning

confidence: 99%

“…These two-stage arguments have also been used by many researchers, such as in [10,26,32,33] for discretizations of nonlinear Schrödinger equations and in [13,24] for discretizations of equations with Burgers nonlinearity. In the case of quasilinear wave equations, error analysis has been presented recently in [11,23,27] for different methods.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Resultsmentioning

confidence: 99%

“…This subsection studies the stability of the ERKN integrator (11). Proof For (w ⊺ ,ẇ ⊺ ) ⊺ = R(h)(v ⊺ ,v ⊺ ) ⊺ , it has been shown in [9] that…”

Section: Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Functionally-fitted energy-preserving integrators for Poisson systems

Wang

2018

Journal of Computational Physics

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“…1 which induces high oscillations in time. In the paper [10] the approach from [9] was extended and in the onedimensional case a quasilinear wave equation with periodic boundary conditions was studied. However, they assume smooth coefficients and high regularity for the analysis.…”

Section: Introductionmentioning

confidence: 99%

On averaged exponential integrators for semilinear wave equations with solutions of low-regularity

Buchholz

Dörich

Hochbruck

2021

Partial Differ. Equ. Appl.

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In this paper we introduce a class of second-order exponential schemes for the time integration of semilinear wave equations. They are constructed such that the established error bounds only depend on quantities obtained from a well-posedness result of a classical solution. To compensate missing regularity of the solution the proofs become considerably more involved compared to a standard error analysis. Key tools are appropriate filter functions as well as the integration-by-parts and summation-by-parts formulas. We include numerical examples to illustrate the advantage of the proposed methods.

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“…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%

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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Trigonometric integrators for quasilinear wave equations

Cited by 18 publications

References 31 publications

Functionally-fitted energy-preserving integrators for Poisson systems

Functionally-fitted energy-preserving integrators for Poisson systems

On averaged exponential integrators for semilinear wave equations with solutions of low-regularity

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

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