Local discontinuous Galerkin methods for nonlinear dispersive equations

Levy, Doron; Shu, Chi‐Wang; Yan, Jue

doi:10.1016/j.jcp.2003.11.013

Cited by 117 publications

(96 citation statements)

References 21 publications

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“…Some numerical solutions of (3.25) are given in [9], [23]. Although (3.24) does not have compacton solutions, it has travelling wave solutions that describe the local non-smooth behavior of the compacton solutions of (3.25) near the 'edges' of the compacton; for example: …”

Section: Compacton Equationsmentioning

confidence: 99%

Short-Time Existence for Scale-Invariant Hamiltonian Waves

Hunter

2006

J. Hyper. Differential Equations

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Section: Compacton Equationsmentioning

confidence: 99%

Short-Time Existence for Scale-Invariant Hamiltonian Waves

Hunter

2006

J. Hyper. Differential Equations

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“…The numerical discretization of the KdV equation can be done by finite differences [63,58,33], finite volumes [6,22], finite elements [3,9], discontinuous Galerkin [39] and, of course, by spectral methods [43,29,65,36]. However, recently the so-called geometrical numerical discretizations have been developed [46,28,40,38,34,35,16].…”

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confidence: 99%

Geometric numerical schemes for the KdV equation

Dutykh

Chhay

Fedele

2013

Comput. Math. and Math. Phys.

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Abstract. Geometric discretizations that preserve certain Hamiltonian structures at the discrete level has been proven to enhance the accuracy of numerical schemes. In particular, numerous symplectic and multi-symplectic schemes have been proposed to solve numerically the celebrated Korteweg-de Vries (KdV) equation. In this work, we show that geometrical schemes are as much robust and accurate as Fourier-type pseudospectral methods for computing the long-time KdV dynamics, and thus more suitable to model complex nonlinear wave phenomena.

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“…We take a uniform mesh h = ∆x = 200/N and compute the solution up to T = 100 using the three stage third order explicit SSP-RK method (3.17) with time step ∆t = T /M. The errors are measured using the discrete scaled norms E 2 h and E ∞ h , [30] …”

Section: Rates Of Convergences Accuracy Testmentioning

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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Local discontinuous Galerkin methods for nonlinear dispersive equations

Cited by 117 publications

References 21 publications

Short-Time Existence for Scale-Invariant Hamiltonian Waves

Short-Time Existence for Scale-Invariant Hamiltonian Waves

Geometric numerical schemes for the KdV equation

Finite volume methods for unidirectional dispersive wave models

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