Browsing by Author "Heemink, Arnold W."

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    Adaptive Stochastic Numerical Scheme in Parallel Random Walk Models for Transport Problems in Shallow Water
    (Elsevier, 2009) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.
    This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing. We start by considering a basic stochastic Euler scheme for integration of the diffusion and drift terms of the SDEs, with a strong order 1 in the strong sense. The errors due to this scheme depend on the location of the pollutant; it is dominated by the diffusion term near boundaries, and by the deterministic drift further away from the boundaries. Using a pair of integration schemes, one of strong order 1.5 near the boundary and one of strong order 2.0 elsewhere, we can estimate the error and approximate an optimal step size for a given error tolerance. The resulting algorithm is developed such that it allows for complete flexibility of the step size, while guaranteeing the correct Brownian behaviour. Modelling pollutants by non-interacting particles enables the use of parallel processing in the simulation. We take advantage of this by implementing the algorithm using the MPI library. The inherent asynchronic nature of the particle simulation, in addition to the parallel processing, makes it difficult to get a coherent picture of the results at any given points. However, by inserting internal synchronisation points in the temporal discretisation, the code allows pollution snapshots and particle counts to be made at times specified by the user.
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    Coloured Noise for Dispersion of Contaminants in Shallow Waters
    (Elsevier, 2009) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.
    In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.
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    Parallel and Distributed Simulation of Sediment Dynamics in Shallow Water Using Particle Decomposition Approach
    (Elsevier, 2008) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.; Verlaan, Martin
    This paper describes the parallel simulation of sediment dynamics in shallow water. By using a Lagrangian model, the problem is transformed to one in which a large number of independent particles must be tracked. This results in a technique that can be parallelised with high efficiency. We have developed a sediment transport model using three different sediment suspension methods. The first method uses a modified mean for the Poisson distribution function to determine the expected number of the suspended particles in each particular grid cell of the domain over all available processors. The second method determines the number of particles to suspend with the aid of the Poisson distribution function only in those grid cells which are assigned to that processor. The third method is based on the technique of using a synchronised pseudo-random-number generator to generate identical numbers of suspended particles in all valid grid cells for each processor. Parallel simulation experiments are performed in order to investigate the efficiency of these three methods. Also the parallel performance of the implementations is analysed. We conclude that the second method is the best method on distributed computing systems (e.g., a Beowulf cluster), whereas the third maintains the best load distribution.
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    A Particle Model for Simulation of Sediment Transport in Coastal Waters
    (2004) Charles, Wilson M.; Heemink, Arnold W.; Verlaan, Martin
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    Stochastic Particle Models for Transport Problems in Coastal Waters
    (2005) Charles, Wilson M.; Heemink, Arnold W.; Van den Berg, E.
    In this paper transport processes in coastal waters are described by stochastic differential equations (SDEs). These SDEs are also called particle models (PMs). By interpreting a Fokker-Planck equation associated with the SDE as an advection diffusion equation (ADE), it is possible to derive the underlying PM which is exactly consistent with the ADE. Both the ADE and the related classical PM do not take into account accurately the short term spreading behaviour of particles. In the PM this shortcoming is due to the driving noise in the SDE which is modelled as a Brownian motion and therefore has independent increments. To improve the behaviour of the model shortly after the release of pollution we develop an improved PM forced by a coloured noise process representing the short-term correlated turbulent velocity of the particles. This way a more accurate and detailed short-term initial spreading behaviour of particles is achieved. For long-term simulations both the improved and classical PMs are consistent with the ADE. However, the improved PM is relatively easier to handle numerically than a corresponding ADE. In this paper both models are applied to a real life pollution problem in the Dutch coastal waters.
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    Variable Time Stepping in Parallel Particle Models for Transport Problems in Shallow Waters
    (2006) Charles, Wilson M.; Van den Berg, E.; Lin, Hai X.; Heemink, Arnold W.
    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.