Abstract

CHAPTER 2. TEMPORAL INTEGRATION 39
met or maximum number of line searches have been met. Then terminate and
take the iteration.
The convergence theorem of the inexact-Newton-Armijo algorithm [19] is as follows
Theorem 2.4. If the function F and the iterates from the inexact-Newton iteration
satisfy:
•{zm} is bounded
•{F −1 (zm)} is bounded
• F ′ is Lipschitz continuous
then the inexact Newton-Armijo iterates will converge to a root of F ,whereinthe
final stages of the iterations, full Newton steps are taken (i.e. σ =1)andtheq-linear
convergence of inexact-Newton’s method returns.
2.6 Preconditioner Development
To make VODEPK’s implicit ODE integrator run efficiently, a preconditioner was
implemented with the GMRES iterative method to speed up the linear solves for the
Newton steps. When implementing either method from VODEPK, the Jacobian of
F is has the form
F ′ (y) =I − ∆tγ ∂G
∂y