A particle method and numerical study of a quasilinear partial differential equation

Roberto Camassa; Pao-Hsiung Chiu; Long Lee; W.-H. Sheu

doi:10.3934/cpaa.2011.10.1503

Article Contents

2011, Volume 10, Issue 5: 1503-1515. Doi: 10.3934/cpaa.2011.10.1503

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A particle method and numerical study of a quasilinear partial differential equation

1.
The University of North Carolina at Chapel Hill, Phillips Hall, CB #3250, Chapel Hill, NC 27599-3250
2.
Nuclear Engineering Division, Institute of Nuclear Energy Research, Taoyuan County, 32546
3.
Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036
4.
Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei City 106

Received: March 31, 2009

Revised: May 31, 2010

Published: August 2011

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Abstract

We present a particle method for studying a quasilinear partial differential equation (PDE) in a class proposed for the regularization of the Hopf (inviscid Burger) equation via nonlinear dispersion-like terms. These are obtained in an advection equation by coupling the advecting field to the advected one through a Helmholtz operator. Solutions of this PDE are "regularized" in the sense that the additional terms generated by the coupling prevent solution multivaluedness from occurring. We propose a particle algorithm to solve the quasilinear PDE. "Particles" in this algorithm travel along characteristic curves of the equation, and their positions and momenta determine the solution of the PDE. The algorithm follows the particle trajectories by integrating a pair of integro-differential equations that govern the evolution of particle positions and momenta. We introduce a fast summation algorithm that reduces the computational cost from $O(N^2)$ to $O(N)$, where $N$ is the number of particles, and illustrate the relation between dynamics of the momentum-like characteristic variable and the behavior of the solution of the PDE.

Keywords:

Quasilinear partial differential equation,
particle method,
Helmholtz operator,
regularization,
inviscid Burgers,
Hopf equation.

Mathematics Subject Classification: Primary: 65M25; Secondary: 35G25.

Citation:

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