The Particle Method For A Series Of Nonlinear Equations

In recent years, particle methods have become one of the most widespread and useful tools for approximating the solutions of partial differential equations in a variety of fields, which have been developed very well both in theory and practice. The so-called particle method is represented by a collection of particles, which was located in points()ix t and carrying masses()ip t. Equations of evolution in time are then written to describe the dynamics of the location of the particles and their weights. Due to the Lagrangian nature of the method, small scales that might develop in a solution can be easily described with a relatively small number of particles.In this paper, our purpose is to establish the optimal error estimate of the particle method for a series of nonlinear evolutionary partial differential equations, including Camassa-Holm equation, Degasperis-Procesi equation and 2D Euler-Poincare equation. Our particle method is an approximation of the equation in Lagrangian representation X(ξ, t), p(ξ, t), where X(ξ, t), p(ξ, t) represent their points and masses through an appropriate transformation, respectively. Then, we calculate the numerical integration involving kernel1 | |()2 2() expxG xα α-= to obtain the particle solution. Because the solution may be not smooth or unstable, or occur leapt situation, so we make the numerical solution smooth by a mollified function ρ(x)??, which can improve the accuracy of the particle solution. We also prove that the method can be expected to be first order accurate for mollifier label ???? and second order accurate for mesh size h, that is2 O( +h??).In the end, we apply our method to solve some C-H, D-P and E-P equations. Compare the numerical solution(before and after mollified) with the exact solution to obtain their 1l-norm. Then, we test our method though certain numerical results to analysis and prove the accurate order of our method.