A fractional step – exponentially fitted hopscotch scheme for the Streeter‐Phelps equations of river self‐purification

Abstract

This paper describes a numerical approximation scheme for the one‐dimensional Streeter‐Phelps equations of river self‐purification. Our formulation of the problem includes second order kinetics in a coordinate system moving with the river thus analytically accounting for convection. These equations are linearized by using fractional time steps. The effects of reaeration and deoxygenation are accommodated by exponential fitting. The discrete equations are then marched forward in time using the hopscotch scheme which is explicit yet unconditionally stable (albeit conditionally consistent). Numerical examples both with and without dispersion are presented which indicate that the proposed method is much more efficient than a brute force numerical approach. Specifically, the proposed explicit scheme is amenable to parallel implementation.