Volume 26, Issue 1
Capturing Near-Equilibrium Solutions: A Comparison Between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes

Maria Han Veiga, David A. Velasco-Romero, Rémi Abgrall & Romain Teyssier

Commun. Comput. Phys., 26 (2019), pp. 1-34.

Published online: 2019-02

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  • 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.

  • Keywords

Numerical methods, benchmark, well-balanced methods, discontinuous Galerkin methods.

  • AMS Subject Headings

65M60, 65Z05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-26-1, author = {}, title = {Capturing Near-Equilibrium Solutions: A Comparison Between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes}, journal = {Communications in Computational Physics}, year = {2019}, volume = {26}, number = {1}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0071}, url = {http://global-sci.org/intro/article_detail/cicp/13023.html} }
TY - JOUR T1 - Capturing Near-Equilibrium Solutions: A Comparison Between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes JO - Communications in Computational Physics VL - 1 SP - 1 EP - 34 PY - 2019 DA - 2019/02 SN - 26 DO - http://dor.org/10.4208/cicp.OA-2018-0071 UR - https://global-sci.org/intro/article_detail/cicp/13023.html KW - Numerical methods, benchmark, well-balanced methods, discontinuous Galerkin methods. AB -

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.

Maria Han Veiga, David A. Velasco-Romero, Rémi Abgrall & Romain Teyssier. (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. doi:10.4208/cicp.OA-2018-0071
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