SEMICIRCULANT PRECONDITIONERS FOR 1ST-ORDER PARTIAL-DIFFERENTIAL EQUATIONS

This paper considers solving time-independent systems of first-order partial differential equations (PDEs) in two space dimensions using a conjugate gradient (CG)-like iterative method. The systems of equations are preconditioned using semicirculant preconditioners. Analytical formulas for the eigenvalues and the eigenvectors are derived for a scalar model problem with constant coefficients. The main problems in constructing and analyzing the numerical methods are caused by the numerical boundary conditions required at the outflow boundaries. It is proved that, when the grid ratio is less than one, the spectrum asymptotically becomes two finite curve segments that are independent of the number of gridpoints. The same type of result for a time-dependent problem has previously been established. For the restarted generalized minimal residual (GMRES) iteration, a slight reduction of the grid ratio from one substantially improves the convergence rate. This is also predicted by an asymptotic analysis of the eigenvalues.