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

Hoel, Hakon A.

Electronic journal of differential equations, Tome 2007 (2007)

Résumé

We consider a numerical scheme for entropy weak solutions of the DP (Degasperis-Procesi) equation

$ 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.

Détail

Classification : 35Q53, 37K10
Keywords: shallow water equation, numerical scheme, entropy weak solution, shockpeakon, shockpeakon collision