On behavior of preconditioned methods for a class of compact finite difference schemes in solution of hyperbolic equations

In this article, We apply Krylov subspace methods in combination of the ADI, BLAGE,... method as a preconditioner for a class of linear systems arising from compact finite difference schemes in solution of hyperbolic equations ??? subject to appropriate initial and Dirichlet boundary conditions, where )u,uu,t,F(x,t)u(x,-u txxxtt ? is constant. We show The BLAGE preconditioner is extremely effective in achieving optimal convergence rates. Numerical results performed on model problem to confirm the efficiency of our approach.