Circulant preconditioners for second order hyperbolic equations (Q1195923)

The authors are concerned with the numerical solution of initial-boundary value problems for linear second order hyperbolic equations. The problems are discretized based on implicit time discretization and central differencing in the space variables with respect to uniform time and space steps. The arising linear systems are solved by preconditioned conjugate gradient methods using circulant preconditioners that preserve the block structure of the coefficient matrix. Denoting by \(\alpha\) the ratio of the time and space steps and by \(m\) the number of grid points in each direction it is shown that the condition number behaves like \(O(\alpha)\) for \(\alpha\ll m\) and \(O(m)\) for \(m\ll\alpha\) wich has to be compared to \(O(\alpha^ 2)\) and \(O(m^ 2)\) for the original matrix. The theoretical findings are supported by numerical results.