An operator splitting method for the Degasperis–Procesi equation

Feng, Bao Feng; Liu, Y.

doi:10.1016/j.jcp.2009.07.022

Cited by 29 publications

(29 citation statements)

References 44 publications

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“…8. This result is similar to the results presented in [7], we can observe that a structured peakon train is generated gradually. Furthermore, we compare the conservative quantities preserved by different methods in Fig.…”

Section: General Initial Value Problemssupporting

confidence: 91%

“…Based on above results, Matsuno obtained the multisoliton solutions of the DP equation for the case κ = 0 [6]. Furthermore, Lundmark and Szmigielski found the explicit form of multipeakon solutions for κ = 0 by solving an inverse scattering problem of a discrete cubic string [7][8][9]. Additionally, the peakon solutions for these two equations are orbitally stable [10].…”

Section: Introductionmentioning

confidence: 99%

“…Coclite, Karlsen and Risebro constructed a set of splitting approximations and proved that they converged to entropy weak solutions [14]. Feng and Liu developed another different operator splitting method for the DP equation [7], which was based on a second order TVD method and a linearized implicit finite difference method. However, the spatial accuracies of all above methods are no higher than second order.…”

Section: Introductionmentioning

confidence: 99%

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A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

Yunrui Guo,

Wenjing Yang,

Hong Zhang

et al. 2019

AAMM

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The Degasperis-Procesi (DP) equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use an optimized finite difference weighted essentially non-oscillatory (OWENO) scheme. New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation, which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations. For the elliptic equation, the Fourier pseudospectral method (FPM) is employed to discretize the high order derivative. Due to the combination of the WENO reconstruction and FPM, the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions. The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

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Section: General Initial Value Problemssupporting

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

Yunrui Guo,

Wenjing Yang,

Hong Zhang

et al. 2019

AAMM

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“…Important questions of stability and general analytic results dealing with DP peakons and the DP wave breaking have been addressed [15][16][17][18]. A considerable amount of work has been also carried out on adapting numerical schemes to deal with the DP equation; we just mention a few: an operator splitting method of Feng & Liu [19], or numerical schemes discussed by Coclite et al [20] and Hoel [21].…”

Section: Introductionmentioning

confidence: 99%

Colliding peakons and the formation of shocks in the Degasperis–Procesi equation

Szmigielski

Zhou

2013

Proc. R. Soc. A.

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The Degasperis-Procesi equation (DP) is one of several equations known to model important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of the DP equation that these two effects can be studied in an explicit manner with the help of the multi-peakon ansatz. In essence, this ansatz allows one to model wave breaking as a collision of hypothetical particles (peakons and antipeakons), called henceforth collectively multipeakons. It is shown that the system of ordinary differential equations describing DP multi-peakons has the Painlevé property which implies a universal wave breaking behaviour, that multipeakons can collide only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of peakon-antipeakon pairs at different locations. Moreover, it is demonstrated that each peakonantipeakon collision results in creation of a shock thus making possible a multi-shock phenomenon.

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“…Li and Olver [28] established the local well-posedness of the CH equation in the nonhomogeneous Sobolev space H s with s > 3/2. Numerical strategies for solving the CH equation include finite difference methods [10,23], pseudo-spectral methods [27,24], local discontinuous Galerkin method [40], operator splitting methods [5,16], multi-symplectic methods [12,13,44] and other effective methods (e.g., see Refs. [8,17]).…”

Section: Introductionmentioning

confidence: 99%

Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

Jiang

Wang

Gong

2020

Applied Numerical Mathematics

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In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system with a quadratic energy functional, which inherits a modified energy conservation law. The new system are then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, the Runge-Kutta method and the Gauss collocation method are applied for the resulting semi-discrete system to arrive at some arbitrarily high-order fully discrete schemes. We prove that the schemes provided by the Runge-Kutta method with the certain condition and the Gauss collocation method can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.

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An operator splitting method for the Degasperis–Procesi equation

Cited by 29 publications

References 44 publications

A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

Colliding peakons and the formation of shocks in the Degasperis–Procesi equation

Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

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