Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

01 May 2018

In shallow water flow and transport modeling, the monotone upstream-centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second-order accuracy. The most important step in MUSCL-type schemes is the MUSCL reconstruction, which calculate extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during the reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a two-point stencil that uses the cell centers of the neighbor cell and the so-called far-neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes non-trivial. In this work, two novel TVD schemes are developed based on different techniques for calculating the upwind slope and downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to two existing MUSCL reconstruction techniques and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.