Mathematical study of linear morphodynamic acceleration and derivation of the MASSPEED approach

Morphological accelerators, such as the MORFAC (MORphological acceleration FACtor) approach (Roelvink, 2006), are widely adopted techniques for the acceleration of the bed evolution, which reduce the computational cost of morphodynamic numerical simulations. In this work we apply an acceleration to the one-dimensional morphodynamic problem described by the de Saint Venant-Exner model by multiplying all the spatial derivatives related to the mass or momentum flux by an acceleration factor >= 1 which may be different for each equation. The goal is to identify the best combination of the accelerating factors for which (i) the bed responds linearly to hydrodynamic changes; (ii) a decrease of the computational cost is obtained. The sought combination is obtained by studying the behavior of an approximate solution of the three eigenvalues associated with the flux matrix of the accelerated system. This approach allows to derive a new linear morphodynamic acceleration technique, the MASSPEED (MASs equations SPEEDup) approach, and the a priori determination of the highest possible acceleration for a given simulation. In this new approach both mass conservation equations (water and sediment) are accelerated by the same factor, differently from the MORFAC approach where only the sediment mass equation is modified. The analysis shows that the MASSPEED gives a larger validity range for linear acceleration and requires smaller computational costs than that of the MORFAC approach. The MASSPEED approach is then implemented using an adaptive approach that applies the maximum linear acceleration similarly to the implementation of the Courant-Friedrichs-Lewy stability condition. Finally, numerical simulations have been performed in order to assess accuracy and efficiency of the new approach. Results obtained in the long-term propagation of a sediment hump demonstrate the advantages of the new approach. The validation of the method is performed under steady or quasi-steady flow conditions, whereas further investigation is needed to extend morphological accelerators to fully unsteady flows.