Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation

The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

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Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

Monthly Weather Review ◽

10.1175/mwr-d-12-00303.1 ◽

2013 ◽

Vol 141 (9) ◽

pp. 3037-3051 ◽

Cited By ~ 17

Author(s):

Paul D. Williams

Keyword(s):

Nonlinear System ◽

Third Order ◽

First Order ◽

Time Stepping ◽

The Third ◽

Numerical Integrations ◽

Seventh Order ◽

Ocean Models ◽

Stability And Accuracy ◽

Fifth Order

Abstract The leapfrog time-stepping scheme makes no amplitude errors when integrating linear oscillations. Unfortunately, the Robert–Asselin filter, which is used to damp the computational mode, introduces first-order amplitude errors. The Robert–Asselin–Williams (RAW) filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. However, it has not previously been shown how to further improve the accuracy by eliminating the third- and higher-order amplitude errors. Here, it is shown that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It is further shown that the use of a more discriminating (1, −4, 6, −4, 1) filter instead of a (1, −2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy. The proposed new schemes are tested in numerical integrations of a simple nonlinear system. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many atmosphere and ocean models.

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Explicit Discrete Dispersion Relations for the Acoustic Wave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes

Advanced Structured Materials - Computer Methods in Mechanics ◽

10.1007/978-3-642-05241-5_1 ◽

2010 ◽

pp. 3-17 ◽

Cited By ~ 4

Author(s):

Mark Ainsworth ◽

Hafiz Abdul Wajid

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Spectral Element ◽

Dispersion Relations ◽

Acoustic Wave Equation

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Design of Runge-Kutta based split-explicit time integration algorithms for the NEMO ocean model

10.5194/egusphere-egu21-9078 ◽

2021 ◽

Author(s):

Nicolas Ducousso ◽

Florian Lemarié ◽

Gurvan Madec ◽

Laurent Debreu

Keyword(s):

Time Integration ◽

Internal Gravity Waves ◽

Positive Definiteness ◽

Ocean Model ◽

Splitting Technique ◽

Time Stepping ◽

Phase Errors ◽

Runge Kutta ◽

Leapfrog Scheme ◽

Stability And Accuracy

<p>The NEMO ocean model is currently based on the Leapfrog scheme that provides a good combination between simplicity and efficiency for low-resolution global simulations. However, this scheme is subject to difficulties that question its relevance at high-resolution : the necessary damping of its computational mode, e.g. via a Robert-Asselin filter, affect stability and increases amplitude and phase errors of the physical mode ; because it is unconditionally unstable for diffusive processes, monotonicity or positive-definiteness comes at a substantial cost and complication. The evolution toward a 2-level time stepping algorithm based on Runge-Kutta schemes is studied. Special attention is given to how to articulate a mode-splitting technique to handle the fast dynamics associated with the free surface. Linear stability analyses of several Runge-Kutta based, split-explicit algorithms are performed and the most promising ones are identified. They allow a good compromise between robustness, stability and accuracy for integration of internal gravity waves, Coriolis and advection processes. Idealized test-cases illustrate the benefits associated to the revised time-stepping compared to the original Leapfrog.</p>

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Time-stepping evolution of wave equation using a Laguerre polynomial scheme

10.1190/segam2015-5879725.1 ◽

2015 ◽

Author(s):

Eduarda C. G. Rego* ◽

Reynam C. Pestana ◽

Edvaldo Araujo

Keyword(s):

Wave Equation ◽

Laguerre Polynomial ◽

Time Stepping ◽

Polynomial Scheme

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The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations

Monthly Weather Review ◽

10.1175/2010mwr3601.1 ◽

2011 ◽

Vol 139 (6) ◽

pp. 1996-2007 ◽

Cited By ~ 53

Author(s):

Paul D. Williams

Keyword(s):

Small Time ◽

Time Stepping ◽

Phase Errors ◽

Power Series Expansions ◽

Analytic Expressions ◽

Optimal Values ◽

The Stability ◽

Stability And Accuracy ◽

Total Model ◽

Elastic Pendulum

Abstract Errors caused by discrete time stepping may be an important component of total model error in contemporary atmospheric and oceanic simulations. To reduce time-stepping errors in leapfrog integrations, the Robert–Asselin–Williams (RAW) filter was proposed by the author as a simple improvement to the widely used Robert–Asselin (RA) filter. The present paper examines the behavior of the RAW filter in semi-implicit integrations. First, in a linear theoretical analysis, the stability and accuracy are interrogated by deriving analytic expressions for the amplitude errors and phase errors. Then, power-series expansions are used to interpret the leading-order errors for small time steps and hence to identify optimal values of the filter parameters. Finally, the RAW filter is tested in a realistic nonlinear setting, by applying it to semi-implicit integrations of the elastic pendulum equations. The results suggest that replacing the RA filter with the RAW filter could reduce time-stepping errors in semi-implicit integrations.

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