Stability and convergence of trigonometric integrator pseudospectral discretization for N-coupled nonlinear Klein–Gordon equations

Dong, Xuanchun

doi:10.1016/j.amc.2014.01.144

Cited by 14 publications

(24 citation statements)

References 15 publications

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“…In this section, we state our error bounds for the trigonometric integrator (7) and its fully discrete version (19) when applied to the quasilinear wave equation (1).…”

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

“…We will universally require Assumptions 1-3 on the filter functions of the trigonometric integrator (7). In addition, we will require that the exact solution u(x, t) to (1) satisfies the following assumption.…”

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

“…Remark 3.3. For small nonlinearities with |κ| 1, we need only Assumptions 1 and 2 on the filter functions of the trigonometric integrator (7) to prove the global error bound, as explained in Remark 2.1. In this case, the necessary energy estimates can be proved and bounded in a simpler fashion and the underlying ellipticity of the second order operator is more easily proved on long time scales.…”

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

“…where Λ s (·) is a continuous non-decreasing function. Throughout the proof of Theorem 3.2, we make use of the fact that the numerical flow ϕ τ given by (7) maps H 2 × H 1 to itself and more generally H s+1 × H s to itself for s ≥ 1, as stated in the following lemma. This property of an explicit numerical method is in the quasilinear case by no means natural.…”

Section: Basic Estimatesmentioning

confidence: 99%

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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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“…In this section, we state our error bounds for the trigonometric integrator (7) and its fully discrete version (19) when applied to the quasilinear wave equation (1).…”

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

Section: Statement Of Global Error Boundsmentioning

confidence: 99%

Section: Basic Estimatesmentioning

confidence: 99%

See 2 more Smart Citations

Trigonometric integrators for quasilinear wave equations

Gauckler

Marzuola

et al. 2018

Math. Comp.

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“…ERKN integrators for integrating (5) were first formulated in [44], and in this paper we consider only one-stage explicit scheme.…”

Section: Erkn Integratorsmentioning

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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