One-dimensional Euler equation algorithm with a very high asymptotic convergence rate (Q1115720)

An algorithm is devised which solves the steady one-dimensional Euler equations by an iteration scheme. The algorithm uses residuals computed from the conservation variables in place of residuals based on Riemann variables. The unsteady Euler equations are used as the model for deriving the correction at each iteration level. The residuals, computed using two-point windward differences, are combined according to characteristic equations to produce the corrections. The stagnation enthalpy is taken to be its final steady value at all points and at all iterations, resulting in characteristic equations different from the usual ones for 1-D unsteady flow. Because the corrections are based on the individual characteristic equations, maximum values of the two local time steps, one corresponding to the forward wave and one corresponding to the backward wave, can be used. Errors propagate a full mesh interval at each calculation. By sweeping first in the forward direction and then in the backward and using the most recent values of the variables, extremely high convergence rates are achieved. Reduction of the residuals by 12 orders of magnitude is achieved in three or less cycles for a flow with no shock waves in a constant area duct. For flows with area change and shock waves, this number rises to about 12 cycles. Nonreflecting wave boundary conditions are used. No artificial dissipation, damping or multigrid acceleration is employed. Simple variations of the program can be employed to show that the maximum achievable convergence rates using only a single local time step based on the forward wave are much slower.