A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms☆

By Laurent Gosse. 2000

223 citations recorded by the source corpus.

Abstract We propose a way to construct robust numerical schemes for the computations of numerical solutions of one- and two-dimensional hyperbolic systems of balance laws. In order to reduce the computational cost, we selected the family of flux vector splitting schemes. We reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations. This is applied to a 1D shallow water system with topography and to a 2D simplified model of two-phase flows with damping effects. Numerical results and comparisons with a classical centered discretizations scheme are supplied.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/0f86b7bfd36c618220d2df6f1681625534c284b1

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Tsunami modelling with adaptively refined finite volume methods* (2011, methodology): “…for such systems (see e.g. Bale, LeVeque, Mitran and Rossmanith (2002), Bouchut (2004), George (2008), Greenberg and LeRoux (1996), Botta, Klein, Langenberg and Lützenkirchen (2004), Gallardo, Parés and Castro (2007), Gosse (2000), LeVeque (2010) and Noelle, Pankrantz, Puppo and Natvig (2006)).”
Well-Balanced High-Order Centred Schemes for Non-Conservative Hyperbolic Systems. Applications to Shallow Water Equations with Fixed and Mobile Bed (2009, methodology): “Numerical treatment of the friction term is made by using the approach of Gosse [16,17], which leads to the following system of governing equations:”
A Survey of High Order Schemes for the Shallow Water Equations (2014, background): “A class of first order accurate flux-vector-splitting schemes based on the theory of nonconservative products was proposed in [24].”
Sediment transport models in Shallow Water equations and numerical approach by high order finite volume methods (2008, unknown): “See [50], [4], [26], [27], [40], [22], [23], [46], [9], [56], [47], [12], [13], [3], [6], [43].”
Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws (2002, methodology): “This formulation has been already used in several satisfactory numerical approaches, see [26, 24 , 27] and justified rigorously in the scalar case, [23].”
Numerical methods for nonconservative hyperbolic systems: a theoretical framework (2006, unknown): “preserve some equilibria: see [2], [3], [5], [7], [10], [11], [12], [17], [18], [23], [24], [27], [29], [31], [32], [36], [37], [38], [39], .”
Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows (2006, background): “Greenberg, LeRoux and coworkers developed schemes based on the solution of the non-homogeneous Riemann-problem, see [13,11,10].”
A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows (2004, background): “Several schemes have been proposed that satisfy this property, [23], [17], [13], [11], [30], [29], [3], but the difficulty is then to get schemes that also satisfy very natural properties such as conservativity of the water height h, nonnegativity of h, the ability to compute dry states h = 0 and tr”
Godunov method for nonconservative hyperbolic systems (2007, unknown): “1) (see [8–11, 16]).”
Convergence of the Upwind Interface Source Method for Hyperbolic Conservation Laws (2003, background): “Quite recently, these kinds of numerical processing have been extended to hyperbolic systems of balance laws (like the Saint-Venant system for shallow waters), to obtain stable schemes which preserve the steady states (see [8], [13],[14] and [28], for instance).”
On the well-balance property of Roe?s method for nonconservative hyperbolic systems (2004, background): “Following an idea introduced in [17, 19, 20], a system of conservation laws with source terms or balance law”