213 1999 Henrik Brandén and Sverker Holmgren henrik@tdb.uu.se Convergence acceleration for hyperbolic systems using semicirculant approximations
Abstract
The iterative solution of systems of equations arising from systems of
hyperbolic, time-independent partial differential equations (PDEs)
is studied. The PDE is discretized using a finite volume or finite difference
approximation on a structured grid. A convergence acceleration technique
where a semicirculant approximation of the spatial difference operator is
employed as preconditioner is considered. The spectrum of the preconditioned
coefficient matrix is analyzed for a model problem. It is shown that,
asymptotically, the time step for the forward Euler method could be chosen as
a constant, which is independent of the number of grid points and the
artificial viscosity parameter. By linearizing the Euler equations around an
approximate solution, a system of linear PDE with variable coefficients is
formed. When utilizing the semicirculant (SC) preconditioner for this problem,
which has properties very similar to the full nonlinear equations, numerical
experiments show that the favorable convergence properties hold also here. We
compare the results for the SC method to those of a multigrid (MG) scheme. The
number of iterations and the arithmetic complexities are considered, and it is
clear that the SC method is more efficient for the problems studied. Also, the
MG scheme is sensitive to the amount of artificial dissipation added, while
the SC method is not.