Computing Invariant Solutions of PDEs with Symmetries

Friday, April 24, 2009 - 1:25pm - 2:25pm
Vincent 570
Vanessa Lopez (IBM)
We consider the problem of numerically computing solutions of evolutionary nonlinear partial differential
equations (PDEs) with
a finite-dimensional group of symmetries. Specifically, we look for solutions that are fixed by elements of
the equations' symmetry
group. The latter class includes time-periodic solutions. We work with the complex Ginzburg-Landau equation
in one space dimension, which has a 3-parameter group of symmetries generated by space-time translations and
a rotation
of the (complex) amplitude. The spectral-Galerkin method used to discretize the PDE will be described, along
with the approach
for solving the resulting system of nonlinear algebraic equations which allowed us to identify multiple new
solutions in a chaotic
region of the CGLE.

Due to the relatively small number of unknowns considered (2,000 - 3,000 after discretization), it was
possible to use a direct
method for linear systems as part of the process for solving the nonlinear system. However, for problems
with a large
number of unknowns, iterative methods for linear systems are required. We will conclude our talk with a
discussion on the use
of such methods for solving these types of problems.