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


Veiga, Maria Han; Velasco-Romero, David A; Abgrall, Rémi; Teyssier, Romain (2019). Capturing near-equilibrium solutions: a comparison between high-order Discontinuous Galerkin methods and well-balanced schemes. Communications in Computational Physics, 26(1):1-34.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
07 Faculty of Science > Institute for Computational Science
Dewey Decimal Classification:530 Physics
Scopus Subject Areas:Physical Sciences > Physics and Astronomy (miscellaneous)
Language:English
Date:July 2019
Deposited On:08 Mar 2019 11:14
Last Modified:29 Jul 2020 10:19
Publisher:Global Science Press
ISSN:1815-2406
OA Status:Closed
Publisher DOI:https://doi.org/10.4208/cicp.OA-2018-0071

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