Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

Cohen, David E.; Hairer, Ernst; Lubich, Christian

doi:10.1007/s00211-008-0163-9

Cited by 122 publications

(138 citation statements)

References 16 publications

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“…Since the approach via modulated Fourier expansions does not use nonlinear coordinate transforms, it turns out to be applicable also to numerical discretizations of (1), as is shown in a companion paper to the present article [7].…”

Section: Introductionmentioning

confidence: 89%

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Cohen

Hairer

Lubich

2007

Arch Rational Mech Anal

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100

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Section: Introductionmentioning

confidence: 89%

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Cohen

Hairer

Lubich

2007

Arch Rational Mech Anal

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100

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“…Several new schemes have been proposed and analyzed. Short time error bounds have been shown in [9,11,12,13,14,19,28] while long time properties have been studied in [2,3,4,15,16,17]. All these integrators require the evaluation or approximation of the product of a trigonometric operator function with a vector.…”

Section: Introductionmentioning

confidence: 99%

Rational approximation to trigonometric operators

Grimm

Hochbruck

2008

Bit Numer Math

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Abstract.We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

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“…Based on our results, we will show that the ε-scalability of the EWI-SP method for the KGZ system in the simultaneous high-plasma-frequency and subsonic limit regime is τ = O(ε 2 ) and h = O (1). In addition, we also observe that the EWI-SP method nearly conserves the total energy over a long time in practical computation for the dynamics of the KGZ system, which is a favorable property for numerical time integrators and has been extensively studied for second-order ordinary different equations (ODEs) and dispersive partial differential equations (PDEs) in the literatures; see, e.g., [4,5,10,11,12] and references therein.…”

Section: E(t)mentioning

confidence: 76%

“…Table 4.4). Specifically, ECFD-SP conserves the energy in the discretized level and the fluctuation of the energy in the EWI-SP method is bounded for a very long time and decreases quadratically when the time step τ decreases, i.e., the EWI-SP method conserves the energy essentially for the KGZ system in practical computation [12,18,19,21,22]. (vi) Based on these observations, when ε = O(1) and γ = O(1), all the numerical methods perform similarly.…”

Section: Numerical Resultsmentioning

confidence: 99%

An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

Bao¹,

Dong²,

Zhao³

2013

SIAM J. Sci. Comput.

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Abstract. An exponential wave integrator sine pseudospectral method is presented and analyzed for discretizing the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters 0 < ε ≤ 1 and 0 < γ ≤ 1 which are inversely proportional to the plasma frequency and the speed of sound, respectively. The main idea in the numerical method is to apply the sine pseudospectral discretization for spatial derivatives followed by using an exponential wave integrator for temporal derivatives in phase space. The method is explicit, symmetric in time, and it is of spectral accuracy in space and second-order accuracy in time for any fixed ε = ε 0 and γ = γ 0 . In the O(1)-plasma frequency and speed of sound regime, i.e., ε = O(1) and γ = O(1), we establish rigorously error estimates for the numerical method in the energy space H 1 × L 2 . We also study numerically the resolution of the method in the simultaneous high-plasma-frequency and subsonic limit regime, i.e., (ε, γ) → 0 under ε γ. In fact, in this singular limit regime, the solution of the KGZ system is highly oscillating in time, i.e., there are propagating waves with wavelength of O(ε 2 ) and O(1) in time and space, respectively. Our extensive numerical results suggest that, in order to compute "correct" solutions in the simultaneous high-plasma-frequency and subsonic limit regime, the meshing strategy (or ε-scalability) is time step τ = O(ε 2 ) and mesh size h = O(1) independent of ε. Finally, we also observe numerically that the method has the property of near conservation of the energy over long time in practical computations.

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Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations

Cited by 122 publications

References 16 publications

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Long-Time Analysis of Nonlinearly Perturbed Wave Equations Via Modulated Fourier Expansions

Rational approximation to trigonometric operators

An Exponential Wave Integrator Sine Pseudospectral Method for the Klein--Gordon--Zakharov System

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