Analytical and experimental convergence results are presented for a novel pseudo-unsteady solution method for higher-order accurate upwind discretizations of the steady Euler equations. Comparisons are made with an existing pseudo-unsteady solution method. Both methods make use of nonlinear multigrid for acceleration and nested iteration for the fine-grid initialization. The new method uses iterative defect correction. Analysis shows that it not only has better stability but it also has better smoothing properties. The analytical results are confirmed by numerical experiments, which show better convergence and efficiency.