High order TVD scheme for hyperbolic conservation laws

Abstract

In this article, a new third order finite difference scheme for solving initial value problems for conservation laws is introduced. The advantages of this scheme are: it is a third order accuracy in space and time, is simple to implement, it has the lowest order of dissipation which reduces the oscillations generated by the numerical methods. The stability of the new scheme is proved for initial boundary value problems for linear and nonlinscalar problems. Also, the new scheme is reformulated to be total variation diminishing (TVD) i.e., oscillations free. For the nonlinear systems of equations, the extension of the new scheme is presented. Many numerical examples are presented and compared with the exact solutions and other methods.