Time step restrictions for well-balanced shallow water solutions in non-zero velocity steady states

Abstract

Numerical models based on the conservative formulation of the shallow water equations in finite volumes have recently paid special attention to the convenience of a unified discretization of bed slope source terms and pressure momentum fluxes in order to ensure a correct representation of both in steady states. In cases of steady shallow water flow with non-zero velocity, the discrete balance must include the friction term and the good equilibrium of the discrete solution must be revisited. Besides, there is the question of stability. The presence of relatively large friction terms reduces considerably the stability region of the explicit schemes (with either separate or unified discretization). Implicit formulations strongly help to relax the friction related stability restrictions but are only feasible in the context of separated discretization. Therefore, a strategy to blend both approaches (unified explicit and separated implicit) is presented that ensures both a perfect balance in steady state and numerical stability in unsteady cases in the presence of friction terms.