Further Development of a Conservative Front-Tracking Method for Systems of Conservation Laws in One Space Dimension

In this paper, we further analyze, improve and develop the conservative front-tracking method developed in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703). The method in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703) is only first-order accurate near the tracked discontinuities when applied to systems of conservation laws. This is because in the system case near tracked discontinuities there are waves in other characteristic fields and they may propagate across the discontinuities. The method in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703) treats this propagate-across of waves in a first-order fashion. In this paper, we develop a high-order treatment of this wave propagate-across on tracked discontinuities and thus enhance the accuracy of the method. We present a rigorous analysis of truncation error to show that the method equipped with the developed treatment of wave propagate-across is high-order accurate in a certain sense. We also discuss some matters concerning the application of the method to the Euler system of gas dynamics, such as the treatment of reflection wall boundaries, application to the multifluid flows, and the programming of the algorithm. Numerical examples are presented to show the efficiency of the method.