Integration of the EPDiff equation by particle methods

Abstract

Abstract. The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

Elastic collisions among peakon solutions for the Camassa–Holm equation

“…For a more detailed description on the particle method for (1.1), we refer the reader to [6,10]. We begin by searching for a weak solution in the form of a linear combination of Dirac delta distributions.…”

“…The latter can be done in the sense of measures on R. Namely, we choose (x i (0), p i (0)) in such a way such that for any test function φ(x) ∈ C ∞ 0 (R), we have that 6) where…”

“…For a more detailed description of the following properties including their proofs, we refer the reader to [6,10].…”

“…Solutions of (1.1), (1.2) can be accurately captured by using a particle method, as shown in [4][5][6]. In the particle method, described in [6,10], the solution is sought as a linear combination of Dirac distributions, whose positions and coefficients represent locations and weights of the particles, respectively.…”

“…In the particle method, described in [6,10], the solution is sought as a linear combination of Dirac distributions, whose positions and coefficients represent locations and weights of the particles, respectively. The solution is then found by following the time evolution of the locations and the weights of these particles according to a system of ODEs obtained by considering a weak formulation of the problem.…”