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 --> infinity) of the solution of a time-dependent problem. Implicit schemes are then often chosen for their favourable stability properties, permitting large timesteps for efficiency. An important class of implicit schemes involving first- and second-order accurate upwind discretisations is considered. In the limit of an infinite timestep, 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 two-dimensional cases. These model problems are analyzed in detail by both Fourier and matrix analyses. The convergence behaviour appears to be strongly dependent on a parameter beta 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 pseudoconvective (or Fourier) phase can be distinguished, and then a faster asymptotic phase. The extreme parameter values beta = 0 (no upwinding) and beta = 1 (full second-order upwinding) both appear as special cases for which the convergence behaviour degenerates. They are not recommended for practical use. For the intermediate values of beta the pseudo-convection phase is less significant. Fromm's scheme (beta = 1/2) or van Leer's third-order scheme (beta = 1/3) show quite satisfactory convergence behaviour. In the last section experiments for the steady Euler equations are discussed. Comments are given on how well phenomena, understood for the scalar linear model problem, are recognised for this system of more complex nonlinear equations.