Higher-order accurate space-time schemes for computational astrophysics-Part I: finite volume methods

Area of Science:

• Computational astrophysics
• Numerical methods
• Fluid dynamics

Background:

• Computational astrophysics demands high-accuracy numerical schemes beyond traditional fluid dynamics.
• Specialized methods are needed to ensure robustness, positivity of density/pressure, and sub-luminal relativistic flows.

Purpose of the Study:

• To review advanced numerical schemes for high-accuracy computational astrophysics.
• To focus on computer-implementable techniques rather than theoretical underpinnings.

Main Methods:

• Discussion of weighted essentially non-oscillatory (WENO) schemes.
• Analysis of discontinuous Galerkin (DG) schemes.
• Examination of PNPM schemes, balancing accuracy and timestep size.
• Inclusion of strong stability preserving Runge-Kutta and ADER schemes for matched spatial-temporal accuracy.

Main Results:

• WENO schemes offer higher-order extensions to finite volume methods.
• DG schemes provide superior accuracy by evolving all solution moments.
• PNPM schemes present a compromise, allowing larger timesteps.
• ADER and Runge-Kutta schemes enable matched spatial and temporal accuracy.

Conclusions:

• The reviewed schemes (WENO, DG, PNPM) address the unique algorithmic needs of computational astrophysics.
• Effective implementation requires matching spatial and temporal accuracy using methods like ADER and Runge-Kutta.
• These advanced numerical techniques are crucial for advancing computational astrophysics as a precision science.