A General Deterministic Treatment of Derivatives in Particle Methods
A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection-diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing derivatives near the edge of particle coverage is introduced, using "one-sided" integrals that only look for information where it is available. The use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using Fourier methods. Finally, simple tests are performed to demonstrate the characteristics of the treatment, including an assessment of the effects of particle dispersion, and their results are discussed.
© 2002 Elsevier. Received 26 April 2001, revised 25 March 2002, available online 8 August 2002. The first author gratefully acknowledges support under a NSF Graduate Research Fellowship. This research was supported in part by the National Science Foundation under Grant 9501349.