Title: Higher Order Accurate Space-Time Schemes for Computational Astrophysics -- Part I -- Finite Volume Methods,
Abstract: As computational astrophysics comes under pressure to become a precision
science, there is an increasing need to move to high accuracy schemes for
computational astrophysics. Hence the need for a specialized review on higher
order schemes for computational astrophysics.
The focus here is on weighted essentially non-oscillatory (WENO) schemes,
discontinuous Galerkin (DG) schemes and PNPM schemes. WENO schemes are higher
order extensions of traditional second order finite volume schemes which are
already familiar to most computational astrophysicists. DG schemes, on the
other hand, evolve all the moments of the solution, with the result that they
are more accurate than WENO schemes. PNPM schemes occupy a compromise position
between WENO and PNPM schemes. They evolve an Nth order spatial polynomial,
while reconstructing higher order terms up to Mth order. As a result, the
timestep can be larger.
Time-dependent astrophysical codes need to be accurate in space and time.
This is realized with the help of SSP-RK (strong stability preserving
Runge-Kutta) schemes and ADER (Arbitrary DERivative in space and time) schemes.
The most popular approaches to SSP-RK and ADER schemes are also described.
The style of this review is to assume that readers have a basic understanding
of hyperbolic systems and one-dimensional Riemann solvers. Such an
understanding can be acquired from a sequence of prepackaged lectures available
from this http URL. We then build on this
understanding to give the reader a practical introduction to the schemes
described here. The emphasis is on computer-implementable ideas, not
necessarily on the underlying theory, because it was felt that this would be
most interesting to most computational astrophysicists.