Path-conservative positivity-preserving well-balanced finite volume WENO method for porous shallow water equations

Abstract

The present study constructed a path-conservative high-order positivity-preserving well-balanced finite volume Riemann solver for the one-dimensional porous shallow water equations with discontinuous porosity and bottom topography. First, the finite difference equations were formulated using the path-conservative approach to treat non-conservative products in source terms. As the solution depends on the paths, the family of paths was formulated for the solutions to satisfy the mass and energy conservations over the smooth paths connecting the discontinuous geometry (porosity and/or bottom topography). Such a property was achieved using the stationary wave, a stationary weak solution of the porous shallow water equations. Second, to increase computational accuracy, the weighted essentially non-oscillatory (WENO) and the Runge-Kutta methods were implemented for the spatiotemporal discretization of the porous shallow water equations, respectively. Third, a positivity-preserving limiter was employed to provide robustness and efficiency in handling wet and dry problems. Finally, all procedures in the present model were formulated to ensure the well-balanced property (i.e., the exact conservation of initial still water conditions over the irregular geometry). A set of numerical experiments was performed to verify that the present model exhibits positivity-preserving and well-balanced properties along with high-order accuracy and the shock-capturing ability for all types of Riemann problems, including eight wet cases and five dry cases. The numerical results of all cases agree with the analytical solutions.