Large-scale computer simulations are
increasingly crucial in explaining astrophysical phenomena. Just one recent
example is the stunning agreement of the full 3-d solution of Einstein's equations
for colliding black holes with the observed signal from LIGO. For the past 50
years, the dominant computer method for solving these kinds of equations has
remained essentially unchanged. To keep up with continuing advances in
observation, simulations will require more fidelity and higher accuracy. One might think that with exascale machines
becoming available in the next 5 years, this will be easy. This is not true: I
will explain why Moore's Law is broken, and how the next generation of
supercomputers will instead get their power by having millions of processors. Current
codes will not be able to use these machines efficiently. I will describe new
methods for harnessing the power of such exascale computers to solve some of
the largest problems in astrophysics and other areas of science.