A semi–discrete high resolution scheme for nonlinear scalar conservation laws

The purpose of this paper is twofold. Firstly we carry out an extension of the fully discrete third order TVD scheme, for linear case, presented in [8] to nonlinear scalar hyperbolic conservation laws for one and two dimensions. Secondly, we propose a semi-discrete version of the scheme. Time evolution is carried out by the third order TVD RungeKutta method. The advantages of the scheme are its simplicity, third order, non-oscillatory and that can be used for large time steps which can save more time. Examples and convergence rates are presented for the Burger equation for one and two dimensions which confirm the high resolution content of the proposed schemes. We use exact solutions and other methods to validate the results.