Experiments with implicit upwind methods for the Euler equations (Q1069788)

A number of implicit integration schemes for the one-dimensional Euler equations with conservative upwind spatial differencing are tested on a problem of steady discontinuous flow. Fastest convergence (quadratic for the first-order ''backward Euler'' scheme) is obtained with the upwind switching provided by the second author's differentiable split fluxes [in: Lect. Notes Phys. 170 (1982; Zbl 0543.76001) on pp. 507-512], which easily linearize in time. With \textit{P. L. Roe}'s [J. Comput. Phys. 43, 357-372 (1981; Zbl 0474.65066)] nondifferentiable split flux-differences the iterations may get trapped in a limit-cycle. This also happens in a second-order scheme with split fluxes, if the matrix coefficients arising in the implicit time-linearization are not properly centered in space. The use of second-order terms computed from split fluxes degrades the accuracy of the solution, especially if these are subjected to a limiter for the sake of monotonicity preservation. Second-order terms computed from the characteristic variables perform best.