The third-order relaxation schemes for hyperbolic conservation laws

A class of semi-discrete third-order relaxation schemes are presented for relaxation systems which approximate systems of hyperbolic conservation laws. These schemes for the scalar conservation law are shown to satisfy the property of total variation diminishing (TVD) in the zero relaxation limit. A third-order TVD Runge–Kutta splitting method is developed for the temporal discretization of the semi-discrete schemes. Numerical results are given illustrating these schemes on one-dimensional nonlinear problems.