Artificial Viscosity Technique: A Riemann-Solver-Free Method for 2D Urban Flood Modelling on Complex Topography

Abstract

This study deals with a simulation of two-dimensional urban flood problems on complex topography based on a cell-centred finite volume (CCFV) scheme. Unlike many numerical models that use a Riemann solver to deal with discontinuities due to the rapid change of a flow regime (caused by very shallow water) or due to wet–dry problems, an artificial viscosity technique is used in this study to tackle numerical instabilities caused by such discontinuities. This technique is a Riemann-solver-free method for solving the shallow water equations and is constructed from a combination of a Laplacian and a biharmonic operator, in which the variable scaling factor is devised by using the spectral radius of the Jacobian matrix. For a time discretisation, the Runge–Kutta fourth-order scheme is then used to achieve high-order accuracy. In order to avoid a computational overhead, this Runge–Kutta scheme is applied in its hybrid formulation, in which the artificial viscosity is only computed once per time step. Another advantage of our technique is a simple computation of the convective flux which is performed only by averaging the left and right states of every edge instead of evaluating complex if-then-else statements as required in the Riemann solver such as Harten-Lax-van Leer-contact (HLLC) scheme. Other improvement aspects address both the proper treatment of the friction source term when dealing with very shallow water on very rough beds and an advanced wet–dry technique which is solely applied in an edge-based fashion. Our results show that this artificial viscosity technique is highly accurate for solving the shallow water equations. Also, we show that this technique is cheaper than the HLLC scheme and entails a much less computational complexity.