A Local Discontinuous Galerkin Method for KdV Type Equations

Yan, Jue; Shu, Chi‐Wang

doi:10.1137/s0036142901390378

Cited by 301 publications

(225 citation statements)

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“…The local discontinuous Galerkin (LDG) method is an extension of the DG method for solving higher order PDEs. It was first designed for convection-diffusion equations [12], and has been extended to other higher order wave equations, including the KdV equation [25,[37][38][39] and the Camassa-Holm equation [35], see also the recent review paper [36] on the LDG methods for higher order PDEs. The idea of the LDG method is to rewrite higher order equations into a first order system, and then apply the DG method on the system.…”

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confidence: 99%

A local discontinuous Galerkin method for the Burgers–Poisson equation

Liu

Ploymaklam

2014

Numer. Math.

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] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The L 2 -stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method.

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confidence: 99%

A local discontinuous Galerkin method for the Burgers–Poisson equation

Liu

Ploymaklam

2014

Numer. Math.

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“…We use a first order local discontinuous Galerkin method (LDG) with 100 elements in space [26,15] and an IMEX scheme in time [1], with the linear terms treated implicitly and the nonlinear term explicitly. We set the tolerance for POD and EIM to be 10 −13 and 10 −8 respectively.…”

Section: Allan-cahn Equation: Nonlinear Sourcementioning

confidence: 99%

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

Chen

Hesthaven

Zhu

2014

Reduced Order Methods for Modeling and Computational Reduction

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We propose a modified parallel-in-time -parareal -multi-level time integration method which, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built from the information obtained from the fine solver at each iteration. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method for similar problems and the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation methods (EIM), we develop this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate the advantages through algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.

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“…In this section we study numerically this special class of solutions. Recently, a discontinuous Galerkin method was employed to study the same problem [51] the BBM equation under consideration, since the coefficient γ is different. A fine grid with ∆x = 0.001 is required to observe this phenomenon.…”

Section: 4mentioning

confidence: 99%

Finite volume methods for unidirectional dispersive wave models

Dutykh

Katsaounis

Mitsotakis

2012

Numerical Methods in Fluids

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Abstract. We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.

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A Local Discontinuous Galerkin Method for KdV Type Equations

Cited by 301 publications

References 20 publications

A local discontinuous Galerkin method for the Burgers–Poisson equation

A local discontinuous Galerkin method for the Burgers–Poisson equation

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

Finite volume methods for unidirectional dispersive wave models

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