Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Coclite, Giuseppe Maria; Karlsen, Kenneth H.; Risebro, Nils Henrik

doi:10.1093/imanum/drm003

Cited by 54 publications

(42 citation statements)

References 23 publications

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“…Fig. 10(a) and (b) shows the computed solution up to t ¼ 1:3, which agrees with the numerical results in [9]. It is checked here that m 0 ðxÞ ¼ u 0 ðxÞ À u 0;xx ðxÞ changes the sign from positive from negative at x % AE1:22.…”

Section: Wave Breakingsupporting

confidence: 84%

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An operator splitting method for the Degasperis–Procesi equation

Feng¹,

Liu

2009

Journal of Computational Physics

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Section: Wave Breakingsupporting

confidence: 84%

“…A triple collision: a triple collision among two symmetric peakon-antipeakons and one stationary shock peakon were studied theoretically by Lundmark in [36] and numerically by Coclite et al in [9]. The initial condition for this case is uðx; 0Þ ¼ e Àjxþ5j þ sgnðxÞe Àjxj À e ÀjxÀ5j :…”

Section: Interactions For Peakons and Shockpeakonsmentioning

confidence: 99%

An operator splitting method for the Degasperis–Procesi equation

Feng¹,

Liu

2009

Journal of Computational Physics

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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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“…One of the important features of Eq. (1.2) is that it has not only peakon solitons [16,46], but also shock waves [6,32,21].…”

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confidence: 99%

Initial boundary value problems of the Degasperis-Procesi equation

Escher

Yin

2008

Parabolic and Navier–Stokes Equations

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Abstract. We mainly study initial boundary value problems for the Degasperis-Procesi equation on the half line and on a compact interval. By the symmetry of the equation, we can convert these boundary value problems into Cauchy problems on the line and on the circle, respectively. Applying thus known results for the equation on the line and on the circle, we first obtain the local well-posedness of the initial boundary value problems. Then we present some blow-up and global existence results for strong solutions. Finally we investigate global and local weak solutions on the half line and on a compact interval, respectively. One interesting result is that the corresponding strong solution to the Degasperis-Procesi equation on the half line blows up in finite time provided the initial potential, assumed nonpositive, is not identically zero. Another one is that all global strong solutions to the Degasperis-Procesi equation on a compact interval blow up in finite time.2000 Mathematics Subject Classification: 35G25, 35L05. Key words and phrases: Degasperis-Procesi equation, initial boundary value problems, local well-posedness, blow-up and global existence, global and local weak solutions.

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Numerical schemes for computing discontinuous solutions of the Degasperis Procesi equation

Cited by 54 publications

References 23 publications

An operator splitting method for the Degasperis–Procesi equation

An operator splitting method for the Degasperis–Procesi equation

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

Initial boundary value problems of the Degasperis-Procesi equation

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