A localized spatiotemporal particle collocation method for long-time transient homogeneous diffusion analysis

This short communication proposes a novel localized spatiotemporal particle collocation method (LSPCM) to analyze long-time transient homogeneous diffusion problems. A time-dependent general solution of diffusion equation without singularity or near singularity at origin is developed. The LSPCM bypasses Laplace transformation or finite difference method to treat the time derivative of the diffusion equation, and can directly approximate the transient homogeneous diffusion problems. Each discrete node in computational domain is only supported by its nearby nodes covered by the local spatiotemporal subdomain. The physical variable of each discrete node is approximated by the linear combination of the physical variable of its nearby nodes and the time-dependent general solutions. Through satisfying boundary conditions and continuity conditions, the LSPCM eventually results in a large-scale sparse linear system which is easy to store and solve. Numerical experiments show that the LSPCM can accurately simulate long-time transient homogeneous diffusion problems by using a very long-time interval.