According to the ENO scheme on structured grids, a class of weighted ENO finite volume scheme on unstructured mesh is developed. On every control volume, it constructs a new weighted quadratic reconstruction polynomial which can save computational costs. It also uses a method which can resolve the overdetermined systems and do not affect the accuracy of the schemes. Besides, the selection of interpolation points and the construction of weight are presented, Third order TVD Runge Kutta time discretization is used. In order to accelerate the convergence, local time step is introduced. The numerical experiments show the scheme effective.

It presents a class of the second order accurate explicit Gauss schemes with staggered grids for the computation of solutions of single hyperbolic conservation laws in two dimernsions, these schemes are Riemann solver-free and Maximum and Minimum Bounds under the restriction of CFL,and have been extended to system of hyperbolic conservation laws. Because these schemes are constructed under staggered grids and Riemann solver-free,the advantages of these schemes compared to other TVD schemes such as Harten's are:no complete set of eigenvectors is needed and hence weakly hyperbolic system can be solved,faster and programming is much simpler.The numerical solutions obtained in computation Riemann problem are satisfactory.