Convergence Analysis of the Defect-Correction Iteration for Hyperbolic Problems

Abstract

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 \to \infty $) 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...