An implicit form for the Osher upwind scheme

Conservative upwind schemes for the Euler equations, such as the Osher scheme, accurately resolve flow discontinuities and correctly model the physics of the problem. However, these schemes require many more arithmetic operations per integration step than simple central-difference schemes and hence result in large computing times. An implicit version of the first-order- and second-order-accurate Osher schemes in two spatial dimensions and generalized coordinates is developed in this study. Because implicit schemes permit the use of large integration steps, in many cases they require fewer integration steps to reach steady-state (especially in calculations on grids with widely varying mesh-cell sizes). The implicit scheme developed in this study accelerated convergence speeds by almost an order of magnitude in the problems considered. Test cases include quasi-one-dimensional nozzle flow and supersonic flow past a cylinder.