This project is to develop new and improve classical high resolution schemes for the transport simulation in surface water. The terms of advection, diffusion and reaction are solved by coupling the shallow water equations (SWE). Among which, the advection is hard to handle because it is related to the vecocity. These difficulties will increase on unstructured grids in term of inducing numerical diffusion or oscillations.  TVD schemes are investigated to overcome these numerical problems.

The requirements for flux limiter functions preserving TVD are derived based on a 1-d non-uniform grid, and a new TVD region is determined to fit arbitrary 1-d grids. Some second order TVD schemes called improved TVD schemes are developed, such as modified Van Leer scheme, modified Van Albada scheme and modified SUPERBEE scheme. Then they are extended to 2-d untructured grids.
However, the impovement of these TVD scheme on 2-d is still necessary and is underway.

The flow fied is obtained by solving SWE in Riemann solver, The extendation of the high order Riemann solvers such as Roe approximate solver, HLL solver, HLLC solver on unstructured grids is studied too in this project.