Localization effects and measure source terms   in  numerical schemes for balance laws

Gosse, Laurent

doi:10.1090/s0025-5718-01-01354-0

Cited by 42 publications

(53 citation statements)

References 38 publications

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“…In the special subcase which is presented in the introduction of a single equilibrium, it is not completely original, and combined with a Riemann solver it can be interpreted as a well balanced scheme following denominations and ideas of Greenberg et al [11]. Then a proof of convergence relying on much stronger assumptions can be found in Gosse [10]. More generally it falls in a class which has been advocated recently by several authors (see Greenberg et al, for source terms which only depend on x, Gosse and Leroux [9] for sources depending on u only, LeVeque [17], Vazquez-Cendon [26] on upwind discretization of source terms for the Saint-Venant system, Bermudez et al [1] for upwind treatment of sources for 2D shallow water equations on unstructured meshes).…”

Section: Introductionmentioning

confidence: 99%

Equilibrium schemes for scalar conservation laws with stiff sources

Botchorishvili¹,

Perthame

Vasseur

2001

Math. Comp.

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Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

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Section: Introductionmentioning

confidence: 99%

Equilibrium schemes for scalar conservation laws with stiff sources

Botchorishvili¹,

Perthame

Vasseur

2001

Math. Comp.

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“…The aim of this section is to exhibit a sufficient condition on the parameter a, which so far is still not determined, that ensures a discrete entropy inequality of the form 19) which is a discrete counterpart of the energy inequality (2.7) verified by the exact solutions, thus assessing the stability of the method. Here, the numerical entropy flux G(U L , U R ) is to be determined.…”

Section: Non-linear Stabilitymentioning

confidence: 99%

“…The idea of taking a constant-by-cell cross-section goes back to the works of LeRoux and co-workers [21,20] and to the paper of Isaacson and Temple [27], which have been extended by Gosse and co-workers [18,19,1]. The consequence of such a discretisation is to concentrate the source term at the interfaces of the mesh and to ease the construction of well-balanced schemes.…”

Section: Introductionmentioning

confidence: 99%

A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

Coquel

Saleh

Seguin

2014

Math. Models Methods Appl. Sci.

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We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the crosssection and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous with arbitrary large jumps. To do so, we introduce in the first step of the relaxation solver a singular dissipation measure superposed on the standing wave which enables us to control the approximate speeds of sound and thus, the time step, even for extreme initial data.

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“…This problem can be rewritten like in [31], by introducing a steady discontinuous variable a(x) ∈ BV (R), replacing the righthand side of (1.3) by ∓ 1 2λ G(∂ x ϕ; w, z)∂ x a and adding the trivial equation ∂ t a = 0. For discontinuous w, z, this formulation can be unstable because of the products "Heaviside × Dirac"; however, it has been rigorously shown in [20] that these nonconservative products can be rigorously defined as weak limits in the framework of [34] thanks to the uniform BV estimates which come from the linear convection in (1.3) (similar estimates for scalar balance laws are previously given in [15]). …”

Section: Numerical Stiffness and Non-conservative Productsmentioning

confidence: 99%

Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model

Gosse¹

2012

SIAM J. Sci. Comput.

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Abstract. We focus on the numerical simulation of a one-dimensional model of chemotaxis dynamics (proposed by Greenberg and Alt [22]) in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme [16]. We are especially interested in studying the decay onto numerical steady-states for two reasons: 1/ conventional upwind schemes have been shown to stabilize onto spurious non-Maxwellian states (with a very big mass flux, see e.g. [24]) and 2/ the initial data lead to a dynamic which is mostly super-characteristic in the sense of [32] thus the stability results of [16] do not apply. A reflecting boundary condition which is compatible with the well-balanced character is presented; a mass-preservation property is proved and some results on super-characteristic relaxation are recalled. Numerical experiments with coarse computational grids are presented in detail: they illustrate the bifurcation diagrams in [24] which relate the total initial mass of cells with the time-asymptotic values of the chemoattractant substance on each side of the domain. It is shown that the WB scheme stabilizes correctly onto zero-mass flow rate (hence Maxwellian) steady-states which agree with the aforementioned bifurcation diagrams. The evolution in time of residues is commented for every considered test-case.

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Localization effects and measure source terms in numerical schemes for balance laws

Cited by 42 publications

References 38 publications

Equilibrium schemes for scalar conservation laws with stiff sources

Equilibrium schemes for scalar conservation laws with stiff sources

A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model

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