Divergence Form for Bed Slope Source Term in Shallow Water Equations

A novel technique is presented for the treatment of the bed slope source terms within the numerical solution of the shallow water equations. The proposed method consists of writing the bed slope source term as the divergence of a proper matrix, related to the static force due to bottom slope. Such a technique is founded on analytical reasoning and is physically based, so that it can be easily applied to a wide range of numerical models, as it is completely independent of any adopted discretization technique, and requires a minimum computational effort. Herein, we show an application to a Godunov-type model, second order accurate both in space and time, based on the finite-volume method. The presented technique leads to a strong improvement in the source terms numerical treatment, especially for steady states, in which flux gradients are exactly balanced by source terms. A surprising degree of simplicity is maintained, with respect to different existing methods. The numerical model has been applied to a set of classical test cases and to a selected laboratory experiment, in order to verify its stability, accuracy, and applicability to practical real-world cases.