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A new third order finite difference scheme for the solution of initial value problems for hyperbolic conservation laws is presented. The advantages of the scheme are its simplicity, third order accuracy and that it can be used for large time steps which saves more time. The scheme is proved stable for initial and initial boundary value problems for linear case. The technique of making the third order scheme oscillations free (TVD) is carried out. In this paper we extend TVD scheme to two dimension problems. The extension of the TVD scheme to nonlinear system of equations is illustrated by solving shallow water equations. Numerical results are presented and compared with exact solutions and other methods.

Third order TVD scheme for hyperbolic conservation laws