Iterative Solution Methods and Preconditioners for Block-Tridiagonal Systems of Equations
Sverker Holmgren, Kurt Otto · SIAM Journal on Matrix Analysis and Applications · 1992
Systems of equations arising from implicit time discretizations and finite difference space discretizations of systems of partial differential equations in two space dimensions are considered. The nonsymmetric linear systems are solved using a preconditioned CG-like iterative method. A class of preconditioners, referred to as semicirculant, is examined. For a scalar hyperbolic model problem, it is shown that the number of iterations required using a preconditioner in the semicirculant class is independent of the number of unknowns, provided that the quotient $\kappa $ between the time- and space-step is held constant. Also, it is shown that the number of iterations grows no faster than $\sqrt \kappa $. This type of favorable convergence property is also observed in numerical experiments solving more complicated problems.