Power boundedness in the maximum norm of stability matrices
for ADI methods
Severiano González-Pinto, Ernst Hairer and Domingo Hernández-Abreu
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 is the space discretization parameter.
For m≤2 sharper bounds are obtained that are independent of n and N.
Key Words.