On Options for the Numerical Modelling of the Diffusion Term in River Pollution Simulations
In this paper five numerical methods for modelling the diffusion term in a one-dimensional advection-diffusion equation are compared. The motivation behind the work is to find a computationally efficient method for modelling diffusion for incorporating in a semi-Lagrangian approach for advection. Three of the schemes are traditional Eulerian implicit methods (backward, Crank-Nicholson, optimised time-weighted): the other two are based on work by Teixeira (Teixeira, 1998) who proposed a method based on a discrete view of diffusion.
Numerical tests were undertaken in a non-advective uniform flow and concerned the evolution of an initial Gaussian spatial distribution of a conservative solute. Tests covered a range of initial spatial resolutions and a range of dimensionless diffusion numbers. For each test, the number of time steps required until the numerical solution agreed to within a specified tolerance of the exact solution to the problem was noted.
The results showed that an optimized time-weighted implicit scheme (used with a weighting coefficient taking values close to 0.7) was twice as efficient as the backward implicit scheme. Teixeira’s original method proved to be rather inefficient in general, but a modified form of it was as efficient as the optimum time-weighted scheme. The Crank-Nicholson scheme was the most efficient unless grid-scale oscillations were present, when it became very inefficient.
KeywordsDiffusion Term Implicit Scheme Large Time Step Random Walk Theory Diffusion Number
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