High order TVD scheme for hyperbolic conservation laws

In this paper, we introduced a new third order finite difference scheme for solving initial value problems for hyperbolic conservation laws. This method is a finite difference scheme combining a dissipative scheme (diffusion scheme) and a dispersive scheme (leap frog scheme) to get a dispersive scheme with less spurious oscillations. The new scheme has the following advantages: it is a third order accuracy in space and time, simple to implement, it has the lowest order of dissipation which reduces the spurious oscillations generated by the numerical methods. The new scheme is proved stable for initial boundary value problems for linear and nonlinear scalar problems. The total variation diminishing (TVD) technique of making the third order scheme oscillations free is carried out. The extension of the scheme to to nonlinear systems of equations is presented. To validate the performance of the proposed scheme, a numerical experiments with one-and two-dimensional problems are presented and compared with exact solutions and other methods.