Optimal Smoothing in Function-Transport Particle Methods for Diffusion Problems

✍ Scribed by Aaron L. Fogelson; Robert H. Dillon

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 368 KB
• Volume: 109
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1993.1208

✦ Synopsis

We discuss a class of particle methods for diffusion problems with small diffusivity, in which diffusion is modeled by random walk update of the particle positions; each particle carries a point-value of the problem's initial data; and the numerical solution is obtained as a discrete convolution of the particle data with an approximate (\delta)-function. While it is widely believed that such a particle method fails to converge, we prove that if the number of particles (M) and the degree of smoothing, measured by the width (\epsilon) of the approximate (\delta)-function, are coupled so that ((M \epsilon)^{-1} \rightarrow 0) as (M \rightarrow \infty) and (\epsilon \rightarrow 0), then the computed solution (u_{M}^{\epsilon}(x, t)) converges to the solution (v(x, t)) of the diffusion equation in the sense that (E\left(\left|u_{M}^{\epsilon}(x, t)-v(x, t)\right|^{2}\right) \rightarrow 0). We also present numerical results which illustrate the theory, and, in particular, show that the method performs uniformly well in the limit that the diffusion coefficient vanishes. (c) 1993 Academic Press. Inc.