A new nnd difference scheme of second-order in time and space

The study by H. X. Zhang shows that in order to suppress the spurious oscillation at both upstream and downstream of the shock, the coefficient of the third-order derivative on the right hand side of the modified equation of the difference scheme must be positive upstream and negative downstream of the shock. According to this principle, a new non-oscillatory, containing no free parameters and dissipative difference scheme of second-order both in time and space is proposed. It is proved that this scheme possesses TVD property and is generalized Gudunov scheme of second-order. In the presence of the shock wave in the flow field, this scheme is the generalization and improvement of the Lax-Wendroff scheme. Several numerical examples are given which demonstrate that the proposed scheme is non-oscillatory of high order accuracy and high resolution. It also has the advantages of compact form, greater maximum allowable Courant number and convenient to use.