Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Gosse, Laurent; Vauchelet, Nicolas

doi:10.1137/151004653

Cited by 16 publications

(30 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A convenient way to make it proper is to compute, in the discretization, the velocity and the density at the same grid points. This fact has already been noticed in [36,30] and is also illustrated numerically in Section 6.…”

Section: Resultssupporting

confidence: 75%

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Delarue¹,

Lagoutìère²,

Vauchelet³

2020

Annales Henri Lebesgue

Self Cite

View full text Add to dashboard Cite

A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order 1/2 in Wasserstein distance. The paper is illustrated by numerical simulations that indicate that this convergence order should be optimal.We will denote the right-hand side (and thus the left-hand side as well) by M 1,∆x .(ii) Bound on the second moment. There exists a constant C > 0, independent of the parameters of the mesh, such that, for all n ∈ N * ,where we recall that t n = n∆t.By definition of the macroscopic velocity (2.15) and by (2.16), we also have where we exchanged the role of J and K in the latter sum. We deduce that it vanishes. Thus,(ii) For the second moment, still using (2.14) and a similar discrete integration by parts, we getAs a consequence of Lemma 3.1, we have |a i n J | ≤ w ∞ . Using moreover the mass conservation, we deduce that the last term is bounded by w ∞ ∆t d i=1 ∆x i . Moreover, applying Young's inequality and using the mass conservation again, we getWe deduce then that there exists a nonnegative constant C only depending on d and w ∞ such thatMoreover using the definition of α L in (4.26), we computex J ρ n J .By conservation of the center of mass, see Lemma 3.2 (i), we deduce that

show abstract

Section: Resultssupporting

confidence: 75%

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Delarue¹,

Lagoutìère²,

Vauchelet³

2020

Annales Henri Lebesgue

Self Cite

View full text Add to dashboard Cite

show abstract

“…The present work somehow completes the former ones [26,30,31] where hydrodynamic limits, involving finite-time concentrations, were considered; hereafter, diffusive limits yielding smooth solutions…”

supporting

confidence: 70%

“…Accordingly, general properties of eigenfunctions for each of the stationary kinetic models are stated in Appendix, along with a new result on exponential monomials, see [38].Remark 1.2 (Notations) When u ∈ R N and v ∈ R M , the matrix u ⊗ v is an element of M N ×M whose coefficients are (u k v ℓ ). We will also commonly use the abuse of notations 1 1 + u ⊗ v ∈ M N ×M , with coefficients 1 1 + u k v ℓ k,ℓ .The present work somehow completes the former ones [26,30,31] where hydrodynamic limits, involving finite-time concentrations, were considered; hereafter, diffusive limits yielding smooth solutions…”

mentioning

confidence: 60%

Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting

Gosse

Vauchelet

2019

Numer. Math.

Self Cite

View full text Add to dashboard Cite

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced method applied to such kinetic equations leads to time-marching schemes involving a "scattering S-matrix", itself derived from a normal modes decomposition of the stationary solution. One common feature these models share is the type of diffusive approximation: their macroscopic densities solve drift-diffusion systems, for which a distinguished numerical scheme is Il'in/Scharfetter-Gummel's "exponential fitting" discretization. We prove that the well-balanced schemes relax, within a parabolic rescaling, towards the Il'in exponential-fitting discretization by means of an appropriate decomposition of the S-matrix. This is the so-called asymptotic preserving (or uniformly accurate) property.Obviously, such a general statement contains several former ones, among which the two-stream Goldstein-Taylor model relaxing to the heat equation, [29], or the so-called "Cattaneo model of chemotaxis", [22]; some of these results were surveyed in [24, Part II].Yet, as a guideline for more involved calculations, we first explain in §3 how the limiting process works for a simple two-stream approximation of (1.1), the so-called "Greenberg-Alt" model of chemotaxis [32]. We mention that for this simple two-velocity model, a numerical scheme formulated in terms of (2.10) with 2 × 2 S-matrices is provided in both [24, page 158] and [30, Lemma 4.1] (for the purpose of hydrodynamic limits, though). Another type of closely related "diffusive limit" involving a 2 × 2 S-matrix was studied in [27].This elementary calculation carried out on the two-stream "Greenberg-Alt" model (3.18) reveals why it is rather natural to expect that a well-balanced algorithm (3.24) (based on stationary solutions) may relax, within a parabolic scaling, toward the exponential-fit scheme (3.21)-(3.22) for the corresponding asymptotic Keller-Segel model. However, as our Theorem 1.1 covers also continuous-velocity models discretized with general quadrature rules, we present in §2 our strategy of proof: in particular, the general scheme involving a scattering matrix is presented in (2.10) and the importance of the decomposition of the scattering matrix (2.11) is emphasized. In §4, such a strategy is applied to the simplest case of continuous equation, namely the "grey radiative transfer" model (4.29). For this system, it is shown in Theorem 4.6 that our numerical scheme relaxes to the finite-difference discretization of the heat equation (4.55). In §5, the case of the Othmer-Alt [44] model of chemotaxis dynamics (5.56)-(5.57) is handled in a similar manner (at the price of more intricate computations, though), yielding asymptotically the scheme (5.59), this is Theorem 5.5. At last, in §6, the case of a Vlasov-Fokker-Planck model ...

show abstract

“…For example, an even and odd formalism may be used to construct an asymptotic preserving scheme towards the parabolic modified Keller-Segel system [29] or Volpert calculus enables to derive an asymptotic preserving scheme in the hydrodynamic limit [73]. Some well-balanced schemes may be found in [53] or, more recently in [56], where the authors introduce a scheme, which is at the same time well-balanced, following the approach of [53], and asymptotic-preserving using Volpert calculus, as in [73]. Another technique consists in constructing a two-steps scheme , combining a well-balanced step and an asymptotic-preserving scheme [42].…”

Section: Propertiesmentioning

confidence: 99%

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Ribot

2018

ESAIM: ProcS

View full text Add to dashboard Cite

Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.

show abstract

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Cited by 16 publications

References 42 publications

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Some examples of kinetic schemes whose diffusion limit is Il’in’s exponential-fitting

A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Contact Info

Product

Resources

About