Implicit upwind methods for the Euler equations

An investigation is conducted regarding the performance of an implicit method for time-integration of the one-dimensional Euler equations, taking into account the use of conservative upwind-biased spatial differencing for a proper treatment of shocks. It is found that quadratic convergence to a steady state incorporating a shock can be achieved with first-order upwind differencing. A Riemann solver which permits continuous differentiation is needed to obtain fast convergence. Incomplete linearization in time of the implicit scheme may cause the iteration process to end in a limit cycle. The decrease in convergence speed observed for a second-order upwind scheme is found to be compensated by an increase in the accuracy of the solution.