High order finite volume method for solving porous shallow water equations over discontinuous geometry

The present study presents the Riemann solvers for solving porous shallow water equations over the discontinuous porosity and bottom topography. Firstly, extending previous works for solving non-strictly hyperbolic systems, we construct the exact Riemann solver. In which, the stationary wave is considered as the elementary wave reflecting the effects of the standing discontinuity in geometry. In particular, in order to guarantee the uniqueness of the stationary wave, we prove that there are geometric functions to which the monotonic criterion can be applied despite the co-existence of variations in porosity and bottom. From the intensive investigation of the Riemann problem of porous shallow water equations, 17 types of solution structures, 10 wet and 7 dry cases, are confirmed. Secondly, reflecting the structure of the exact Riemann solver, we implement approximate Riemann solvers based on the path-conservative finite volume method. Utilizing the geometric functions which satisfy the monotonic criterion for regularization paths, the non-conservative products are measured in order to reflect the effect of the geometric discontinuity at the cell interface. Furthermore, a Well-balanced WENO reconstruction is implemented to achieve high-order accuracy and exact C-property. Positivity-preserving property is considered to prevent depth from becoming negative during simulation. Stationary wave reconstruction is formulated to reflect the structure of the exact solution. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and convergence to Riemann solutions for all types of wave configurations. The numerical results show good agreements with analytical solutions.