A two-dimensional artificial viscosity technique for modelling discontinuity in shallow water flows

In this study, a two-dimensional cell-centred finite volume scheme is used to simulate discontinuity in shallow water flows. Instead of using a Riemann solver, an artificial viscosity technique is developed to minimise unphysical oscillations. This is constructed from a combination of a Laplacian and a biharmonic operator using a maximum eigenvalue of the Jacobian matrix. In order to achieve high-order accuracy in time, we use the fourth-order Runge–Kutta method. A hybrid formulation is then proposed to reduce computational time, in which the artificial viscosity technique is only performed once per time step. The convective flux of the shallow water equations is still re-evaluated four times, but only by averaging left and right states, thus making the computation much cheaper. A comparison of analytical and laboratory results shows that this method is highly accurate for dealing with discontinuous flows. As such, this artificial viscosity technique could become a promising method for solving the shallow water equations.