Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire

Després, Bruno; Lagoutìère, Frédéric

doi:10.1016/s0764-4442(99)80301-2

Cited by 39 publications

(102 citation statements)

References 2 publications

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“…The construction is an extension on arbitrary grids of the recent work [13,26] about non dissipative TVD schemes on regular grids. We assume that the fluxes α jk have to satisfy a compatibility principle for all k ∈ I + (j), a compatibility principle for all k ∈ I − (j), plus a kind of L ∞ estimate.…”

Section: Some Non Linear Schemesmentioning

confidence: 99%

“…Now we derive the LVD estimate for all schemes (13) such that the L ∞ estimate (24) is true. Necessarily α j is a convex combination of the upper and lower bounds of (24).…”

Section: Longitudinal Variation Diminishing Estimatementioning

confidence: 99%

“…(note that s j and Ω j depend on ∆x), b: for each ∆x, (α ∆x ) n is given by the scheme (13)(14), c: the flux is a convex combination…”

Section: A Proof Of Weak Convergence For Lvd Schemesmentioning

confidence: 99%

“…In Section 3 we propose a 2D generalization on arbitrary grids of TVD schemes, based on an extension of the formalism [13,26]. These schemes are non linear in the general case and satisfy a generalized Harten formalism.…”

Section: Introductionmentioning

confidence: 99%

“…Our proof does not assume such a dissipation process: it is an important advantage of LVD estimates. It leaves place for the study of convergent non linear and non dissipative schemes for linear advection and transport equation on arbitrary grids (recent progress has been made on finite difference grids [13,26]). We delay to a forthcoming work the question of finding optimal non dissipative LVD schemes for "real computations".…”

Section: Introductionmentioning

confidence: 99%

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Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Després

Lagoutìère

2001

ESAIM: M2AN

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Abstract.We study a family of non linear schemes for the numerical solution of linear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate. This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.Résumé. Nousétudions une famille de schémas non linéaires pour l'approximation numérique de l'advection linéaire sur grille quelconque en dimension d'espace supérieureà un. Une preuve de convergence est proposéeà partir d'une estimation de la variation longitudinale. Cette estimation est une généralisation multidimensionnelle discrète de l'estimation TVD discrète, bien connue en dimension un d'espace.Mathematics Subject Classification. 76M12, 65M12.

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Section: Some Non Linear Schemesmentioning

confidence: 99%

“…Now we derive the LVD estimate for all schemes (13) such that the L ∞ estimate (24) is true. Necessarily α j is a convex combination of the upper and lower bounds of (24).…”

Section: Longitudinal Variation Diminishing Estimatementioning

confidence: 99%

“…(note that s j and Ω j depend on ∆x), b: for each ∆x, (α ∆x ) n is given by the scheme (13)(14), c: the flux is a convex combination…”

Section: A Proof Of Weak Convergence For Lvd Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Després

Lagoutìère

2001

ESAIM: M2AN

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A random choice scheme for scalar advection

Gallouët

Hurisse

Kokh³

2023

Numerical Methods in Fluids

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SummaryThis paper is dedicated to a numerical method based on a random choice as proposed in Glimm's scheme. It is applied to the problem of advection of a scalar quantity. The numerical scheme proposed here relies on a fractional step approach for which: the first step is performed using any classical finite‐volume scheme, and the second step is a cell‐wise update. This second step is a projection based on a random choice. The resulting scheme possesses a very low level of numerical diffusion. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh‐size. The algorithm performs very well on one‐dimensional and multi‐dimensional problems. This algorithm is very easy to implement even for multi‐processor computations.

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A Hybrid Finite Volume Method for Advection Equations and Its Applications in Population Dynamics

Yang¹,

Tine

2017

Numerical Methods Partial

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International audienceWe present in this article a very adapted ﬁnite volume numerical scheme for transport type-equation. Thescheme is an hybrid one combining an anti-dissipative method with down-winding approach for the ﬂux(Després and Lagoutière, C R Acad Sci Paris Sér I Math 328(10) (1999), 939–944; Goudon, Lagoutière,and Tine, Math Method Appl Sci 23(7) (2013), 1177–1215) and an high accurate method as the WENO5one (Jiang and Shu, J Comput Phys 126 (1996), 202–228). The main goal is to construct a scheme ableto capture in exact way the numerical solution of transport type-equation without artifact like numericaldiffusion or without “stairs” like oscillations and this for any regular or discontinuous initial distribution.This kind of numerical hybrid scheme is very suitable when properties on the long term asymptotic behaviorof the solution are of central importance in the modeling what is often the case in context of populationdynamics where the ﬁnal distribution of the considered population and its mass preservation relation arerequired for prediction

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Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire

Cited by 39 publications

References 2 publications

Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

A random choice scheme for scalar advection

A Hybrid Finite Volume Method for Advection Equations and Its Applications in Population Dynamics

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