A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation

Karakashian, Ohannes A.; Makridakis, Charalambos

doi:10.1090/s0025-5718-2014-02878-0

Cited by 17 publications

(21 citation statements)

References 31 publications

(53 reference statements)

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“…As the conservative methods for KdV equation [3,14,40], Zakharov system [34], Schrödinger-KdV system [35], short pulse equation [41], etc., various conservative numerical schemes are proposed to "preserve structure". Usually, the conservative schemes can help reduce the phase error along the long time evolution.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

Zhang

Xia

2020

J Sci Comput

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In this paper, we develop discontinuous Galerkin (DG) methods for the Ostrovsky-Vakhnenko (OV) equation, which yields the shock solutions and singular soliton solutions, such as peakon, cuspon and loop solitons. The OV equation has also been shown to have a bi-Hamiltonian structure. We directly develop the energy stable or Hamiltonian conservative discontinuous Galerkin (DG) schemes for the OV equation.Error estimates for the two energy stable schemes are also proved. For some singular solutions, including cuspon and loop soliton solutions, the hodograph transformation is adopted to transform the OV equation or the generalized OV system to the coupled dispersionless (CD) system. Subsequently, two DG schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the DG schemes, including shock solution and, peakon, cuspon and loop soliton solutions.

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Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

Zhang

Xia

2020

J Sci Comput

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“…In particular, mass and momentum are such quantities. In [BCKX13] and [KM15] the authors propose and analyse a dG scheme for generalised KdV equations. The scheme itself is very carefully designed to be conservative, in that the invariant corresponding to the momentum is inherited by the discretisation.…”

Section: Introductionmentioning

confidence: 99%

The design of conservative finite element discretisations for the vectorial modified KdV equation

Jackaman

Papamikos

Pryer

2019

Applied Numerical Mathematics

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We design a consistent Galerkin scheme for the approximation of the vectorial modified Kortewegde Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.

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“…However, this method does not achieve the optimal (k+1)-th order of accuracy when piecewise polynomials of odd degree k is used (this accuracy degeneracy is well known for even k and is also shown to exist for odd k recently in [13]). In [16], the authors developed the L 2 conservative LDG numerical scheme and compared with dissipative LDG scheme for KdV type equations to display the phase error caused by dissipation. Compared with the dissipative scheme, the L 2 conservative in [16] and Hamiltonian conservative LDG numerical scheme proposed in [18] can both reduce the phase error efficiently.…”

Section: Introductionmentioning

confidence: 99%

“…In [16], the authors developed the L 2 conservative LDG numerical scheme and compared with dissipative LDG scheme for KdV type equations to display the phase error caused by dissipation. Compared with the dissipative scheme, the L 2 conservative in [16] and Hamiltonian conservative LDG numerical scheme proposed in [18] can both reduce the phase error efficiently. For even k, these L 2 and Hamiltonian conservative schemes can achieve optimal order of convergence rate, however, for odd k, only reach sub-optimal order.…”

Section: Introductionmentioning

confidence: 99%

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

Zhang¹,

Xia²

2019

CiCP

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In this paper, we develop the Hamiltonian conservative and L 2 conservative local discontinuous Galerkin (LDG) schemes for the Korteweg-de Vries (KdV) type equations with the minimal stencil. For the time discretization, we adopt the semiimplicit spectral deferred correction (SDC) method to achieve the high order accuracy and efficiency. Also we compare the schemes with the dissipative LDG scheme. Stability of the fully discrete schemes is provided by Fourier analysis for the linearized KdV equation. Numerical examples are shown to illustrate the capability of these schemes. Compared with the dissipative LDG scheme, the numerical simulations also indicate that the conservative LDG scheme with high order time discretization can reduce the long time phase error validly.

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A posteriori error estimates for discontinuous Galerkin methods for the generalized Korteweg-de Vries equation

Abstract: Abstract. We construct, analyze and numerically validate a posteriori error estimates for a class of conservative discontinuous Galerkin (DG) schemes for the Generalized Koreteweg-de-Vries (GKdV) equation.

Cited by 17 publications

References 31 publications

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

The design of conservative finite element discretisations for the vectorial modified KdV equation

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

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