Pacific Institute for the Mathematical Sciences - PIMS

Traditional stability concepts for ODE solvers typically deal with
linear equations and/or bounds involving inner-product norms only.
Modern problems of interest are typically nonlinear and in many
casesthe relevant bounds for the problem involve more general
boundedness properties such as positivity or or the total variation
diminishing property. Strong stability preserving (SSP) methods
(referred to also as contractivity preserving, monotonicity preserving,
or total variation diminishing methods) provide such boundedness
properties whenever the desired property is satisfied under forward
Euler integration. After reviewing the relationship between strong
stability preservation and other stability concepts, I will discuss
work on finding SSP Runge-Kutta methods that preserve stability under
the largest possible timestep. Investigation of implicit SSP
Runge-Kutta methods has revealed some remarkable properties, as well as
barriers to their efficiency. This work spurred a new search for
optimal explicit methods, leading to new methods that are preferable to
existing ones in terms of both memory and computational efficiency. I
will also mention some current work on SSP properties of general linear
methods and spectral deferred correction methods.