I will discuss ongoing research on the development of novel methods
for the temporal integration of ODEs and PDEs. The strategy
employed in the numerical methods is based on an iterative deferred
corrections approach, within which we can utilize operator
splitting, implicit-explicit methods, and parallelization in the
temporal direction to achieve better computational efficiency. Much
of the recent work is motivated by fluid-structure interaction
problems in biological systems and I will discuss the difficulties
and progress related to some target applications in this area.