Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study

Meng, Xucheng

doi:10.4208/cicp.oa-2019-0214

Cited by 8 publications

(4 citation statements)

References 45 publications

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“…One of the main interests of water waves with finite depth is the shallow water wave. Recent research focuses of numerical simulations and design of numerical schemes of shallow water waves [2,9,22]. In this section, we also comment on the generalization of our results to finite-depth water waves, not only for shallow water waves.…”

Section: Regarding Water Wave Simulationmentioning

confidence: 85%

On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model

Li¹,

Liu²,

Liu³

et al. 2022

CICP

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Although interest in numerical approximations of the water wave equation grows in recent years, the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In practice of water wave simulations, the tradeoff between efficiency and stability has been a challenging problem. Thus to shed light on the stability condition for simulations of water waves, we focus on a model simplified from the water wave equation of infinite depth. This model preserves two main properties of the water wave equation: non-locality and hyperbolicity. For the constant coefficient case, we conduct systematic stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta methods in time. As a result, an optimal time discretization strategy is provided in the form of a modified CFL condition, i.e. ∆t = O( √ ∆x). Meanwhile, the energy stable property is established for certain explicit Runge-Kutta methods. This CFL condition solves the problem of efficiency and stability: it allows numerical schemes to stay stable while resolves oscillations at the lowest requirement, which only produces acceptable computational load. In the variable coefficient case, the convergence of the semi-discrete approximation of it is presented, which naturally connects to the water wave equation. Analogue of these results for the water wave equation of finite depth is also discussed. To validate these theoretic observation, extensive numerical tests have been performed to verify the stability conditions. Simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

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Section: Regarding Water Wave Simulationmentioning

confidence: 85%

On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model

Li¹,

Liu²,

Liu³

et al. 2022

CICP

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“…In this work we have used an explicit Runge-Kutta scheme where the time step size is subject to the CFL condition (2.32) and can become small for adaptive meshes. Implicit schemes such as the exponential time differencing Runge-Kutta scheme [27] merit future research.…”

Section: Discussionmentioning

confidence: 99%

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

Zhang¹,

Huang²,

Qiu³

2022

CICP

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A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one-and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.

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“…Therefore, it is vital and necessary to design numerical methods to preserve discrete energy dissipation laws. Many efforts have been devoted to the development of numerical methods for energy stability in this active research field, which include, but are not limited to, the convex splitting method [4, 23, 28-30, 33, 45, 54, 56], the average vector field method [6,55], exponential time differencing (ETD) method [10,11,13,22,36,37,44,51] and the invariant energy quadratization (IEQ) method [34,[66][67][68]70]. In addition, the scalar auxiliary variable (SAV) method [1,14,15,27,57,58,62,69] has been successfully developed, inspired by a similar idea of the IEQ method.…”

Section: Introductionmentioning

confidence: 99%

A Scalar Auxiliary Variable (SAV) Finite Element Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System with Dynamic Boundary Conditions

Yao¹,

Wang²

2024

JCM

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In this paper, we consider the Cahn-Hilliard-Hele-Shaw (CHHS) system with the dynamic boundary conditions, in which both the bulk and surface energy parts play important roles. The scalar auxiliary variable approach is introduced for the physical system; the mass conservation and energy dissipation is proved for the CHHS system. Subsequently, a fully discrete SAV finite element scheme is proposed, with the mass conservation and energy dissipation laws established at a theoretical level. In addition, the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

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Localized Exponential Time Differencing Method for Shallow Water Equations: Algorithms and Numerical Study

Cited by 8 publications

References 45 publications

On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model

On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A Scalar Auxiliary Variable (SAV) Finite Element Numerical Scheme for the Cahn-Hilliard-Hele-Shaw System with Dynamic Boundary Conditions

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