Abstract

CHAPTER 2. TEMPORAL INTEGRATION 41
integrate out for a long period of time in hopes of reaching this state from some
initial condition. This method is needlessly costly to perform if we are not interested
in tracking the transient solution. Another method, the one we use, is to apply an
inexact-Newton method to the Wigner equation. We use the FORTRAN nonlinear
solver NITSOL [29] which implements a forward differencing Newton-GMRES(m)
with line searching method. We use K −1 as a preconditioner for the GMRES solve,
since K −1 is an approximate inverse of the Jacobian W ′ ( f). The argument for this
is the same as that for the preconditioner used in the temporal integration.
For this problem, we wanted the initial iterate to satisfy the boundary conditions
given in (1.11), (1.12) to avoid spending iterations resolving them. So for each x in the
domain, we enforced the boundary conditions, which depend only on k. The 2-norm
of the residual of the initial iterate was on the order of 10 −3 and 10 −4 , depending on
the grid.
Tabulated below are the preconditioned and unpreconditioned Newton- GMRES(m)
statistics. For the preconditioned GMRES(m), m = 100 and for the unpreconditioned
problem, we use m = 800. We terminate the nonlinear iteration when the first of
three things happen: 400 nonlinear iterations have been taken, the 2-norm of the
step is less than 10 −13 , or the 2-norm of the function is less than 10 −13 . We also used
ηm = 1 . In the tables below, NLI stands for the number of nonlinear iterations, Avg.
10
Kry/N means the average number of Krylov iterations per Newton, and Line Search
Red. is the number of line search reductions taken in the whole Newton iteration.