Implementation of the Preissmann scheme to solve the Hairsine-Rose erosion equations: Verification and evaluation

Abstract

Erosion and deposition are fundamental surface processes, and the Hairsine-Rose (HR) model describing these dynamic processes for individual particle size classes has increasingly been applied at a range of scales. There are no exact solutions of the HR equations for the dynamic erosion of multi-size-classed soil. For numerical solution, upwind explicit schemes which have been applied so far generally require extremely small time intervals (10−2–10−3s) for numerical stability, thus limiting the computational efficiency. The Preissmann scheme, which is flexible and can be fully implicit, has been widely used to simulate open-channel flows. The objective of the paper was to implement, for the first time, the Preissmann scheme to solve the HR equations, and to verify the accuracy and evaluate the efficiency of the scheme. The paper shows that (1) implementation of the Preissmann scheme, while complex, is straightforward, and the numerical solution of the HR equations can be reduced to a set of 2I algebraic equations where I is the number of size classes, with no matrix operations or iteration involved; (2) the numerical solution is verified for steady state solutions, and for the only exact solution for uniform grain-sized soil known to exist; (3) for two benchmark tests to simulate dynamic erosion of multi-size-classed soil. The time interval using the Preissmann scheme can be much larger (∼1s), and there is improvement in overall computational efficiency so long as the time interval for the Preismann scheme is at least 6 times larger than that for upwind explicit schemes for the case of flow-driven erosion. The improvement in computational efficiency can be even more pronounced for overland flows with rainfall where the water depth can be quite small near the upper end of the slope.