Geodesic
A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation
Electronic Journal of Differential Equations, Tome 2007 (2007).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35Q53, 37K10
Keywords: shallow water equation, numerical scheme, entropy weak solution, shockpeakon, shockpeakon collision 
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     author = {Hoel, Hakon A.},
     title = {A numerical scheme using multi-shockpeakons to compute solutions of the {Degasperis-Procesi} equation},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2007},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a294/}
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Hoel, Hakon A. A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation. Electronic Journal of Differential Equations, Tome 2007 (2007). http://geodesic.mathdoc.fr/item/EJDE_2007__2007__a294/