Capturing near-equilibrium solutions: a comparison between high-order discontinuous Galerkin methods and well-balanced schemes

TL;DR

This paper compares high-order discontinuous Galerkin methods and well-balanced schemes for accurately capturing near-equilibrium solutions, demonstrating their effectiveness through test cases and a realistic protoplanetary disc simulation.

Contribution

It provides a detailed comparison of high-order and well-balanced numerical schemes for equilibrium solutions, including implementation and application to astrophysical scenarios.

Findings

Both methods effectively capture equilibrium states.

High-order schemes reduce truncation errors significantly.

Well-balanced schemes preserve equilibrium exactly.

Abstract

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.