Convergence Analysis of the Defect-Correction Iteration for Hyperbolic Problems

This paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady Euler equations. Such equations are also commonly solved by (pseudo) time integration, the steady solution being achieved as the limit (for t ! 1) of the solution of a time-dependent problem. Implicit schemes are then often chosen for their favourable stability properties, permitting large time-steps for eeciency. An important class of implicit schemes involving rst and second order accurate upwind discretisations, is considered. In the limit of an innnite time-step, these implicit schemes approach defect-correction algorithms. Thus our analysis is informative for both types of construction. Simple scalar linear model problems are introduced for the one-dimensional and the two-dimensional case. These model problems are analyzed in detail, both by Fourier and by matrix analyses. The convergence behaviour appears to be strongly dependent on a parameter that determines the amount of upwinding in the discretisation of the second order scheme. In general, in the convergence behaviour of the iteration, after an impulsive initial phase a slower pseudo-convective (or Fourier) phase can be distinguished, and nally again a faster asymptotic phase. The extreme parameter values = 0 (no upwinding) and = 1 (full second order upwind-ing) both appear as special cases for which the convergence behaviour degenerates. They are not recommended for practical use. For the intermediate values of the pseudo-convection phase is less signiicant. Fromm's scheme (= 1=2) or van Leer's third order scheme (= 1=3) show a quite satisfactory convergence behaviour. In the last section we show experiments for the steady Euler equations. Comments are given on how well phenomena, understood for the scalar linear model problem, are recognised for this system of more complex nonlinear equations.