A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations

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 wellbalanced 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.

Improved Riemann solvers for an accurate resolution of 1D and 2D shock profiles with application to hydraulic jumps

“…To this end, let us define first an approximate flux functionF(x, t) with a similar structure than U(x, t). As in the ARoe solver, left and intercell numerical fluxes F − i and F + i+1 can be defined using (20).…”

“…Such states cannot be removed even when refining the grid, hence numerical schemes must be designed in a particular way to overcome such flaw. The solution is computed using the ARoe scheme in [20], which in this case is reduced to the traditional Roe method as the source term is nil. The solution is presented in Figure 2 x | (x − 80) 2 + (y − 50) 2 ≤ 400 , x ∈ Ω and the bed elevation is set to z(x, y) = 0.…”

“…whereγ are the components ofΓ i+1/2 = P −1 i+1/2 δF i+1/2 , the projection of the jump of the rotated extrapolated fluxes across cell interface, This approach is based on the use of the internal solution for the normal discharge, hu, provided by the ARoe solver for the x-split SWE, to approximate the numerical flux of the shear discharge, huv, which is the third component of F in (11). According to previous studies [20], the internal solution for the discharge, hereafter referred to as star solution and denoted by (hu) * , is constant across the contact discontinuity at x = 0 introduced by the source term, that is (hu) + i+1/2 = (hu) − i+1/2 = (hu) * i+1/2 . Such solution stands for the intercell numerical discharge and can be used to construct the flux of the shear wave using the upwind approach as follows…”

“…The resulting solvers, hereafter referred to as ARoe SWC SR 1 and 2, aim at the accurate resolution of hydraulic jumps over complex bathymetry in the SWE. The novelty of the proposed method is the combination of: (a) a flux extrapolation/correction method that provides the exact solution for the discharge in steady conditions under complex bathymetry, (b) a contact wave smearing method that avoids the carbuncle, (c) an upwinding algorithm that considers the contribution of the source term in the solution of the Riemann Problem (RP) ensuring the well-balanced property [20,19] and (d) an entropy correction method, described in [20].…”

“…The termS i+1/2 stands for a suitable approximation of the integral of the source term inside a control volume defined by [−∆x/2, ∆x/2]. Imposing the consistency condition [20], the following constraint is noticed…”