Variable Time Stepping In Parallel Particle Models For Transport Problems In Shallow Waters
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W. M. Charles, E. van den Berg, H.X. Lin & A. W. Heemink
Stochastic differential equations (SDEs) are stochastic in nature. The SDEs under consideration are often called particle models (PMs). PMs in this article model the simulation of transport of pollutants in shallow waters. The main focus is the derivation and efficient implementation of an adaptive scheme for numerical integration of the SDEs in this article. The error determination at each integration time step near the boundary where the diffusion is dominant is done by a pair of numerical schemes with strong order 1 of convergence and that of strong order 1.5. When the deterministic is dominantwe use the aforementioned order 1 scheme and another scheme of strong order 2. An optimal stepsize for a given error tolerance is estimated. Moreover, the algorithm is developed in such a way that it allows for a completely flexible change of the time stepsize while guaranteeing correct Brownian paths. The software implementation uses the MPI library and allows for parallel processing. By making use of internal synchronisation points it allows for snapshots and particle counts to be made at given times, despite the inherent asynchronicity of the particles with regard to time. Keywords: adaptive schemes, Wiener processes, SDEs, particle model, variable stepsize, parallel computing, speed up.
adaptive schemes, Wiener processes, SDEs, particle model, variable stepsize, parallel computing, speed up.