A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations. (December 2016) - Murillo, J.; Navas-Montilla, A.
Highlights: The complete derivation and description of an approximate Riemann solver for general hyperbolic systems of equations with nonconservative terms is presented. The final updating scheme is presented in both the intercell flux form and in the fluctuation form, including a proper description of the approximate numerical flux based on the definition of RH conditions across each wave and using the inner states. When applied to the shallow water equations, it is shown that the use of well balanced numerical schemes limits the range of application of the numerical solver when moving to realistic scenarios. An energy balance numerical algorithm is derived, where the complete set of source terms is fully exercised under any flow condition involving high slopes and arbitrary shear stress. Positivity conditions are explored under a general framework and numerical simulations can be accurately performed recovering an appropriate selection of the time step, allowed by a detailed analysis of the approximate solver. The use of case-dependent threshold values is unnecessary and exact mass conservation is preserved. Abstract: Powerful numerical methods have to consider the presence of source terms of different nature, that intensely compete among them and may lead to strong spatiotemporal variations in the flow. When applied to shallow flows, numerical preservation of quiescent equilibrium, also known as the well-balanced property, is still nowadays the keystone for the formulation of novel numerical schemes. But this condition turns completely insufficient when applied to problems of practical interest. Energy balanced methods (E-schemes) can overcome all type of situations in shallow flows, not only under arbitrary geometries, but also with independence of the rheological shear stress model selected. They must be able to handle correctly transient problems including modeling of starting and stopping flow conditions in debris flow and other flows with a non-Newtonian rheological behavior. The numerical solver presented here satisfies these properties and is based on an approximate solution defined in a previous work. Given the relevant capabilities of this weak solution, it is fully theoretically derived here for a general set of equations. This useful step allows providing for the first time an E-scheme, where the set of source terms is fully exercised under any flow condition involving high slopes and arbitrary shear stress. With the proposed solver, a Roe type first order scheme in time and space, positivity conditions are explored under a general framework and numerical simulations can be accurately performed recovering an appropriate selection of the time step, allowed by a detailed analysis of the approximate solver. The use of case-dependent threshold values is unnecessary and exact mass conservation is preserved.