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Particle methods for dispersive equations. (English) Zbl 0991.65008
Summary: We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of P. Degond and F.-J. Mustieles [SIAM J. Sci. Stat. Comput. 11, No. 2, 293-310 (1990; Zbl 0713.65090)] to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular, we are interested in nonlinear equations which generate structures that have nonsmooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton-compacton type interaction.

MSC:
65C35 Stochastic particle methods
65Z05 Applications to the sciences
35Q35 PDEs in connection with fluid mechanics
35K55 Nonlinear parabolic equations
76R50 Diffusion
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