On the use of the reach‐back characteristics method for calculation of dispersion

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Abstract
The Holly‐Preissmann two‐point finite difference scheme (HP method) has been popularly used for solving the advection equation. The key idea of this scheme is to solve the dependent variable (i.e. the concentration for the pollutant transport problem) by the method of characteristics with the use of cubic interpolation on the spatial axis. The interpolating polynomials of higher order are constructed by use of the dependent variable and its derivatives at two adjacent grid points. In this paper a new interpolating technique is introduced for incorporation with the Holly‐Preissmann two‐point method. The new method is denoted herein as the Holly‐Preissmann reach‐back method (HPRB) and allows the characteristics to project back several time steps beyond the present time level. Through stability analyses it has been observed that the increase of the reach‐back time step numbers for the characteristics indeed reduces the numerical damping and dispersive phenomena. A schematic model has been constructed to demonstrate the merits of this new technique for the calculation of the pure advection and dispersion equations. Numerical experiments and comparisons with analytical solutions which support and demonstrate this new technique are presented.

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