Power boundedness in the maximum norm of stability matrices for ADI methods

Abstract

The study of convergence of time integrators, applied to linear discretized PDEs, relies on the power boundedness of the stability matrix R. The present work investigates power boundedness in the maximum norm for ADI-type integrators in arbitrary space dimension m. Examples are the Douglas scheme, the Craig–Sneyd scheme, and W-methods with a low stage number. It is shown that for some important integrators ‖ Rn‖ ∞ is bounded in the maximum norm by a constant times min ((ln (1 + n)) m, (ln N) m) , where m is the space dimension of the PDE, and N≥ 2 is the space discretization parameter. For m≤ 2 sharper bounds are obtained that are independent of n and N.