Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation

We present a class of augmented approximate Riemann solvers for the shallow water equations in the presence of a variable bottom surface. These belong to the class of simple approximate solvers that use a set of propagating jump discontinuities, or waves, to approximate the true Riemann solution. Typically, a simple solver for a system of m conservation laws uses m such discontinuities. We present a four wave solver for use with the the shallow water equations—a system of two equations in one dimension. The solver is based on a decomposition of an augmented solution vector—the depth, momentum as well as momentum flux and bottom surface. By decomposing these four variables into four waves the solver is endowed with several desirable properties simultaneously. This solver is well-balanced: it maintains a large class of steady states by the use of a properly defined steady state wave—a stationary jump discontinuity in the Riemann solution that acts as a source term. The form of this wave is introduced and described in detail. The solver also maintains depth non-negativity and extends naturally to Riemann problems with an initial dry state. These are important properties for applications with steady states and inundation, such as tsunami and flood modeling. Implementing the solver with LeVeque’s wave propagation algorithm [R.J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys. 131 (1997) 327–335] is also described. Several numerical simulations are shown, including a test problem for tsunami modeling.