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.