Electron. J. Diff. Eqns., Vol. 2007(2007), No. 100, pp. 1-22.

A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation

Håkon A. Hoel

Abstract:
We consider a numerical scheme for entropy weak solutions of the DP (Degasperis-Procesi) equation $u_t - u_{xxt} + 4uu_x = 3u_{x}u_{xx}+ uu_{xxx}$. Multi-shockpeakons, functions of the form
$$ u(x,t) =\sum_{i=1}^n(m_i(t) -\hbox{sign}(x-x_i(t))s_i(t))e^{-|x-x_i(t)|}, $$
are solutions of the DP equation with a special property; their evolution in time is described by a dynamical system of ODEs. This property makes multi-shockpeakons relatively easy to simulate numerically. We prove that if we are given a non-negative initial function $u_0 \in L^1(\mathbb{R})\cap BV(\mathbb{R})$ such that $u_{0} - u_{0,x}$ is a positive Radon measure, then one can construct a sequence of multi-shockpeakons which converges to the unique entropy weak solution in $\mathbb{R}\times[0,T)$ for any $T>0$. From this convergence result, we construct a multi-shockpeakon based numerical scheme for solving the DP equation.

Submitted June 13, 2007. Published July 19, 2007.
Math Subject Classifications: 35Q53, 37K10.
Key Words: Shallow water equation; numerical scheme; entropy weak solution; shockpeakon; shockpeakon collision.

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Håkon A. Hoel
Centre of Mathematics for Applications, University of Oslo
P.O. Box 1053, Blindern, NO-0316 Oslo, Norway
email: haakonah1@gmail.com
http://www.folk.uio.no/haakonah

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