Evaluation of overshooting errors in particle methods for diffusion by biased global random walk
DOI:
https://doi.org/10.33993/jnaat351-1020Keywords:
overshooting, global random walk, groundwater contaminationAbstract
The adjustment of grid steps which guarantees that particles methods yield no numerical diffusion inevitably induces overshooting errors in the solution of the parabolic partial differential equations with space variable coefficients. In this paper we give an evaluation of the overshooting errors of the "global random walk" algorithm (GRW), a computational efficient method used in simulations for transport in environmental problems. The evaluation is performed by comparisons between the GRW solutions and those of the "biased global random walk" algorithm (BGRW), a cellular automaton, which is computationally more expensive but is also free of overshooting errors. The reference problem was the diffusive transport in a random velocity field, a model for the transport of the contaminant solutes in groundwater. The evaluation reveals that, for an optimum choice of the parameters, GRW results for time intervals of practical interest lie in ranges of acceptable precision, for both the ensemble averaged observables and for their fluctuations.Downloads
References
Doob, J. L., Stochastic Processes, John Wiley & Sons, London, 1953.
Gardiner, C. W. (1985), Handbook of Stochastic Methods (for Physics, Chemistry and Natural Science), Springer, New York. DOI: https://doi.org/10.1007/978-3-662-02452-2
Kitanidis, P. K., Particle-tracking equations for solution of the advection-dispersion equation with variable coefficients, Water Resour. Res., 30(11), 3225-3227, 1994, https://doi.org/10.1029/94wr01880 DOI: https://doi.org/10.1029/94WR01880
Kloeden, P. E. and E, Platten, Numerical solutions of stochastic differential equations, Springer, Berlin, 1995.
SUN, Ne-Z., Mathematical Modeling in Groundwater Pollution, Springer, New York, 1996. DOI: https://doi.org/10.1007/978-1-4757-2558-2
Suciu, N., C. Vamoş, J. Vanderborght, H. Hardelauf and H. Vereecken, Numerical modeling of large scale transport of contaminant solutes using the global random walk algorithm, Monte Carlo Methods and Appl., 10(2), 153-177, 2004, https://doi.org/10.1163/156939604777303235 DOI: https://doi.org/10.1515/156939604777303235
Suciu, N., C. Vamoş, P. Knabner and U. Rüde, Biased global random walk, a cellular automaton for diffusion, in Simulationstechnique, 18th Symposium in Erlangen, September 2005, F. Hülsemann, M. Kowarschik and U. Rüde (eds.), pp. 562-567, SCS Publishing House e. V., Erlangen, 2005.
Suciu N., C. Vamoş, J. Vanderborght, H. Hardelauf and H. Vereecken, Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res., 42, W04409, 2006, https://doi.org/10.1029/2005wr004546 DOI: https://doi.org/10.1029/2005WR004546
Vamoş, C., N. Suciu and H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comp. Phys., 186(2), 527-544, 2003, https://doi.org/10.1016/s0021-9991(03)00073-1 DOI: https://doi.org/10.1016/S0021-9991(03)00073-1
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