The influence of source terms on stability, accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes

Abstract

SUMMARYThe two-dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two-dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second-order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case.The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant-Friedrichs-Lewy (CFL) condition.

A 2D extension of a Large Time Step explicit scheme (CFL>1) for unsteady problems with wet/dry boundaries

“…Figure 1: Sketch of the contributions sent from interface i+1/2 when λ > 0 (a) and when λ < 0 (b) (6), (7). It is worth stressing that non-linear equations involve rarefaction waves and shocks.…”

“…where c = √ g h. Discretizing the equations on a regular mesh of size ∆x by means of the first order upwind explicit scheme, Roe's linearization [11] allows the expression of the differences in the conserved variables and in the source terms across the grid edge i + 1/2 as a sum of waves [7]:…”

“…where the tilde variables represent average values at each edge andẽ m are the linearized eigenvectors of the Jacobian matrix, and have the corresponding eigenvaluesλ m [7]. The coefficientsα m andβ m contain the linearized set of wave strengths and source strengths respectively, that are explicitly written in [8].…”

“…An appropriate discretization of source terms [7,8,9], present in realistic cases, was adopted, ensuring both that the size of the time step does not have to be reduced below that imposed by the standard CFL number limit, and that the well-balanced property is satisfied. Moreover, boundary conditions and a parameter to internally limit the time step size in the presence of large discontinuities or wet/dry fronts were analyzed and proposed in [6].…”

The formulation of internal boundary conditions in unsteady 2‐D shallow water flows: Application to flood regulation

“…In the last decade, the main effort has been put on keeping a discrete balance between flux and source terms in cases of still water, leading to the notion of well-balanced schemes or C property [Vazquez-Cendon, 1999;Toro, 2001;Rogers et al, 2003;Murillo et al, 2007]. Recently, in order to include properly the effect of source terms in the weak solution, augmented approximate Riemann solvers have been presented [George, 2008;Rosatti et al, 2008;Murillo and Garcıa-Navarro, 2010a].…”