Optimal one-stage and two-stage schemes for steady state solutions of hyperbolic equations (Q1801361)

The convergence of linear two-step iterative methods is considered for the matrix problem \(Ax = b\) in the case when \(A\) is positive definite and the eigenvectors form a basis. It is not required that \(A\) be symmetric. The arguments are made in terms of the eigenvalues \(\mu_ j\) of the iteration matrix \(M\). The author does not mention that it may happen that the value of \(n\) may have to be very large before an estimate of the norm of \(M^ n\) by \(\max_ j| \mu_ j|^ n\) is computationally useful. The results are applied to the long-time behavior of a hyperbolic system in which the characteristic curves all go in one direction, as in supersonic gas dynamics.