By François Bouchut

ISBN-10: 3764366656

ISBN-13: 9783764366650

This publication is dedicated to finite quantity tools for hyperbolic platforms of conservation legislation. The accessory is wear the advance of instruments for reading the nonlinear balance of Godunov schemes. ranging from theoretical concerns, the schemes are derived till a truly useful point, assembly a few required gains resembling for instance the remedy of vacuum in gasoline dynamics. when it comes to assets, the overall notions of consistency, order of accuracy and well-balancing are constructed, and utilized to the development of potent schemes.

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This numerical flux is discontinuous when one of the eigenvalues crosses 0. 7 Chapter 2. 212) where ρ ≥ 0, u are assumed to be given solving ∂t ρ + ∂x (ρu) = 0. 213) For smooth solutions, the two equations can be combined to give ∂t φ + u∂x φ = 0. 214) means that φ is simply passively transported with the flow. The functions ρ, u can be thought to be obtained by solving a system of equations that can involve other quantities, but we need not specify how they are obtained for what we explain here.

6 VFRoe method The VFRoe method has been introduced in [24], [38], [39]. 42), but is rather defined directly as the value of the flux at the approximate interface value. 4), B(Y ) = (dY /dU )F (U )(dY /dU )−1 . 202) and with Y (Ul ) + Y (Ur ) . 204) at least if 0 is not an eigenvalue of B(Y ). If this is the case, Y (x/t = 0) is not well defined and some other formula needs to be used, for example by defining Y (0) = (Y (0−) + Y (0+))/2. This numerical flux is obviously consistent. However, it is not possible for this scheme to analyze the preservation of invariant domains and the existence of entropy inequalities.

The construction of solvers that are able to resolve vacuum is a main point for applications to flows in rivers with Saint Venant type equations. 4 Relaxation solvers The relaxation method is the most recent between the ones presented here. It is used in [63], [30], [17], [11], [26]. We follow here the presentation of [27], [18] (see also [78]). 12. 66) where f (t, x) ∈ Rq , and A(f ) ∈ Rq . We assume that this system is also hyperbolic. 68) L A(M (U )) = F (U ). 66), we define U ≡ Lf. 68) this gives the expected value when f is a maxwellian, f = M (U ).

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Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources by François Bouchut


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