Asymptotic preserving trigonometric integrators for the quantum Zakharov system

Baumstark, Simon; Schratz, Katharina

doi:10.1007/s10543-020-00815-2

Cited by 5 publications

(1 citation statement)

References 34 publications

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“…Recently, there has been growing interest in developing accurate and efficient numerical methods for the QZS (1.1). Baumstark and Schratz [5] developed a new class of asymptotic preserving trigonometric integrators for the QZS (1.1), and the scheme converges to the classical ZS in the limit ε → 0 uniformly. Zhang [56] proposed an explicit time-splitting combined with exponential wave integrator Fourier pseudo-spectral method for the QZS (1.1).…”

Section: Introductionmentioning

confidence: 99%

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Zhang¹,

Jiang²

2022

Preprint

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In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) the original system is then rewritten into a new equivalent system which inherits the quadratic energy based on the energy variational principle; (iii) the resulting system is discretized by symplectic Runge-Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time and can preserve the discrete mass and Hamiltonian energy exactly. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, extensive numerical examples are provided to demonstrate the theoretical claims and illustrate the efficiency of our method.

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Section: Introductionmentioning

confidence: 99%

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Zhang¹,

Jiang²

2022

Preprint

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Uniform error bound of a Crank–Nicolson‐type finite difference scheme for Zakharov system in the subsonic limit regime

Wang

Yang

2023

Math Methods in App Sciences

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In this paper, we use the order reduction method to present a Crank–Nicolson‐type finite difference scheme for Zakharov system (ZS) with a dimensionless parameter ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$, which is inversely proportional to the ion acoustic speed. The proposed scheme is proved to perfectly inherit the mass and energy conservation possessed by ZS, while the invariants satisfied by most existing schemes are expressed by two‐level's solution at each time step. In the subsonic limit regime, that is, when 0<ε≪1$$ 0<\varepsilon \ll 1 $$, the solution of ZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in designing numerical methods and establishing the error estimates, especially in the subsonic limit regime. After proving the solvability of the proposed scheme, we use the cut‐off function technique and energy method to rigorously analyze two independent error estimates for the well‐prepared, less‐ill‐prepared, ill‐prepared initial data, respectively, which are uniform in both time and space for ε∈false(0,1false]$$ \varepsilon \in \left(0,1\right] $$ and optimal at second order in space. Numerical examples are carried out to verify the theoretical results and show the effectiveness of the proposed scheme.

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Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Zhang,

Jiang

2023

Adv Comput Math

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Asymptotic preserving trigonometric integrators for the quantum Zakharov system

Cited by 5 publications

References 34 publications

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

Uniform error bound of a Crank–Nicolson‐type finite difference scheme for Zakharov system in the subsonic limit regime

Arbitrary high-order structure-preserving methods for the quantum Zakharov system

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