Asymptotic High-Order Schemes for $2\times2$ Dissipative Hyperbolic Systems

Aregba-Driollet, Denise; Briani, Maya; Natalini, Roberto

doi:10.1137/060678373

Cited by 17 publications

(26 citation statements)

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“…In the case of a semilinear model, obtained from (1.1) by neglecting the drift term (ρu 2 ) x and taking a linear pressure, i.e. γ = 1, Natalini and Ribot [25] have proposed an Asymptotically High Order method [1] adapted to the case of a system with an external source term and set on a bounded interval. This technique increases the accuracy of the scheme for large times and yields a correct asymptotic stabilization near the nonconstant equilibria.…”

Section: Introductionmentioning

confidence: 99%

A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis

Natalini

Ribot

Twarogowska

2014

Communications in Mathematical Sciences

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Abstract. We introduce a numerical scheme to approximate a quasilinear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which properly handles the presence of vacuum and which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system. Key words. Hyperbolic system with source, chemotaxis, stationary solutions with vacuum, finite volume methods, well-balanced scheme.AMS subject classifications. Primary: 65M08; Secondary: 35L60, 92B05, 92C17. IntroductionThe movement of bacteria, cells or other microorganisms under the effect of a chemical stimulus, represented by a chemoattractant, has been widely studied in mathematics in the last two decades (see [18,22,23,26]), and numerous models involving partial differential equations have been proposed. The basic unknowns in these chemotactic models are the density of individuals and the concentrations of some chemical attractants. One of the most considered models is the Patlak-KellerSegel system [21], where the evolution of the density of cells is described by a parabolic equation, and the concentration of a chemoattractant is generally given by a parabolic or elliptic equation, depending on the different regimes to be described and on authors' choices. The behavior of this system is quite well known now: in the one-dimensional case, the solution is always global in time [24], while in two and more dimensions the solutions exist globally in time or blow up according to the size of the initial data; see [6,7] and references therein. However, a drawback of this model is that the diffusion leads alternatively to a fast dissipation or an explosive behavior, and prevents us from observing intermediate organized structures, such as aggregation patterns.In this paper, we consider a quasi-linear hyperbolic system of chemotaxis introduced by Gamba et al. [11] to describe the early stages of the vasculogenesis process, namely the formation of blood vessels networks during the embryonic development. The model forms a hyperbolic-parabolic system for the following unknowns: the density of endothelial cells ρ(x,t), their momentum ρu(x,t), and the concentration φ(x,t)

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Section: Introductionmentioning

confidence: 99%

A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis

Natalini

Ribot

Twarogowska

2014

Communications in Mathematical Sciences

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“…This is what is done, for instance, constructing the so-called "Asymptotic High-Order" (AHOp) methods [1,3]. Considering, e.g., a scalar hyperbolic balance law like…”

Section: An Example Involving Hyperbolic Pdesmentioning

confidence: 99%

On the interplay between asymptotic and numerical methods to solve differential equations problems

Spigler

2014

Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

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“…In this paper, we propose techniques to formalize the rigorous analysis of the longtime convergence. In particular, these techniques lead to the remarkable and unexpected asymptotic behavior of some Strang splitting schemes, which approximate better the solution in longtime than locally predicted, in the spirit of the asymptotic high-order schemes developed by [ADBN08].…”

Section: Introductionmentioning

confidence: 98%

Asymptotic behaviour of splitting schemes involving time-subcycling techniques

Dujardin¹,

Lafitte

2015

IMA J Numer Anal

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This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi-)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small time-step to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in short-time to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toy-models and is illustrated with several numerical experiments on these toy-models as well as on more complex systems.

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Asymptotic High-Order Schemes for $2\times2$ Dissipative Hyperbolic Systems

Cited by 17 publications

References 16 publications

A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis

A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis

On the interplay between asymptotic and numerical methods to solve differential equations problems

Asymptotic behaviour of splitting schemes involving time-subcycling techniques

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