Two-Point Implicit Scheme for Euler Equations

A new two-point implicit scheme for solving the Euler equations was developed, which is second-order accurate in space and time. This is an extension of the MacCormack implicit scheme, and is taking account of the following points; 1. to solve for the physical quantity itself at a new time-step, not for change of the physical quantity in time, 2. to modify the implicit operators for stability so as not to decrease accuracy of the approximation of the first derivative. The resulting method is efficient and easy to program. This scheme was compared with various other schemes for a shock tube problem. No physically irrelevant pressure distributions of the shock wave obtained by the present method were observed. It was confirmed that the present one produced good results for a large CFL number.