  The Scheduled Relaxation Jacobi (SRJ) method is a linear solver algorithm
which greatly improves the convergence of the Jacobi iteration through the use
of judiciously chosen relaxation factors (an SRJ scheme) which attenuate the
solution error. Until now, the method has primarily been used to accelerate the
solution of elliptic PDEs (e.g. Laplace, Poisson's equation) as the currently
available schemes are restricted to solving this class of problems. The goal of
this paper is to present a methodology for constructing SRJ schemes which are
suitable for solving non-elliptic PDEs (or equivalent, nonsymmetric linear
systems arising from the discretization of these PDEs), thereby extending the
applicability of this method to a broader class of problems. These schemes are
obtained by numerically solving a constrained minimization problem which
guarantees the solution error will not grow as long as the linear system has
eigenvalues which lie in certain regions of the complex plane. We demonstrate
that these schemes are able to accelerate the convergence of standard Jacobi
iteration for the nonsymmetric linear systems arising from discretization of
the 1D and 2D steady advection-diffusion equations.
