A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms
By Yulong Xing, Chi-Wang Shu. Communications in Computational Physics, 2006
121 citations recorded by the source corpus.
Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work (31-33), we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and Runge- Kutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.
Subfields: Applied Mathematics, Computational Mechanics
View Paper: https://www.semanticscholar.org/paper/fc0d0ec512023e3b176ac5545e3c91bc2ae2cf88
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In Applied Mathematics
- Alexander Kurganov
- Chi-Wang Shu
- Guergana Petrova
- Yulong Xing
- Jörn Behrens
- Stefan Vater
- N. Beisiegel
- Smadar Karni
In Computational Mechanics
- Alexander Kurganov
- Chi-Wang Shu
- Guergana Petrova
- Yulong Xing
- Jörn Behrens
- Stefan Vater
- N. Beisiegel
- Eleuterio F. Toro
Citations in the corpus
- A Survey of High Order Schemes for the Shallow Water Equations (2014, methodology): “In this section, we recall the positivity-preserving high order well-balanced finite element DG schemes developed by us in [55, 59, 60, 64, 65].”
- A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: One-dimensional case (2015, methodology): “For (2), exact quadrature rules are a basic requirement for well-balancing [30].”
- A SECOND-ORDER WELL-BALANCED POSITIVITY PRESERVING CENTRAL-UPWIND SCHEME FOR THE SAINT-VENANT SYSTEM ∗ (2007, methodology): “The methods that exactly preserve such solutions are called well-balanced, and we refer the reader to [2, 4, 8, 10, 12, 16, 17, 20, 21, 22, 23, 27, 28], where a variety of highorder well-balanced schemes for the Saint-Venant system can be found.”
- High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems (2009, methodology): “A procedure to systematically modify the WENO procedure so that it is still stable and nonoscillatory in the presence of negative weights was developed in [146] and has been used in many later works, such as in [131], where this technique is used to construct high order central WENO schemes, in [127”