December 10, Monday, 1:00pm,  Courant Institute, rm. 1302
David Keyes,  Columbia University,

A Nonlinearly Implicit Manifesto

Many simulations must be followed over time intervals that are very
long compared to the shortest timescales in the system, e.g.,
convective versus acoustic timescales in aerodynamics, ocean turnover
versus gravity wave timescales in climate, plasma discharge versus
Alfven timescales in tokamaks, piston travel versus fast reaction
timescales in internal combustion. Often, the phenomena associated
with the shortest timescales may safely be put in equilibrium on
physical grounds; however, they control the computational timestep if
an explicit method is used, with the result that even weak scaling
cannot be achieved.  Often, as well, one would ideally employ a
high-order timestepping scheme and take relatively large timesteps for
computational economy; however, if operator splitting techniques are
used, the splitting error defeats this purpose. For these and other
reasons, fully implicit methods are increasingly important for the
nonlinear multiscale applications that pace large-scale simulations in
energy, environmental, and other complex systems. The good news is
that advances in solution algorithms, globalization algorithms, and
software that implements them (without necessarily demanding that the
user constructs a full Jacobian) make implicit methods more inviting
than ever. Moreover, we argue that computational challenges on the
immediate horizon -- uncertainty quantification, inverse problems,
multiphysics coupling, etc. -- are most naturally tackled with fully
nonlinearly implicit formulations well in hand.  This talk illustrates
the case for implicit methods with model problems and challenge
problems arising from systems governed by partial differential
equations.