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

Abstract

We consider a numerical scheme for entropy weak solutions of the DP (Degasperis-Procesi) equation ut - uₓₓt + 4uuₓ = 3uₓuₓₓ + uuₓₓₓ. Multi-shockpeakons, functions of the form u(x, t) = ∑ni=1 (mi(t) - sign(x - xi(t))ss(t))e-|x-xi(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 u0 ∈ L1(ℝ) ∩ BV (ℝ) such that u0 - u0,x is a positive Radon measure, then one can construct a sequence of multi-shockpeakons which converges to the unique entropy weak solution in ℝ x [0, T) for any T > 0. From this convergence result, we construct a multi-shockpeakon based numerical scheme for solving the DP equation.