Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 100, pp. 1-22.

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

Author: Hakon A. Hoel (Univ. of Oslo, Norway)

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.