In this paper, we propose a numerical scheme to solve the kinetic model for chemotaxis phenomena. Formally, this scheme is shown to be uniformly stable with respect to the small parameter, consistent with the fluid-diffusion limit (Keller-Segel model). Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.keyword: Asymptotic preserving scheme; Kinetic theory; Micro-macro decomposition; Chemotaxis phenomena MSC: 65M06, 35Q20, 82C22, 92B0 the movement of cells by a "run & tumble" process [6,7]. Cells move along a straight line in the running phase and make reorientation as a reaction to the surrounding chemicals during the tumbling phase. This is the typical behavior that has been observed in experiments. The resulting kinetic equation, with parabolic scaling, reads  where f (t, x, v) denotes the density of cells, depending on time t, position x ∈ Ω ⊂ R d and velocity v ∈ V ⊂ R d . T is an operator, which models the change of direction of cells and ε is a time scale which here refers to the turning frequency. The function S(t, x) is the chemical concentration, where n denotes the density of cells, and is given byStarting with the kinetic equation (2), one can (at least formally) derive the macroscopic limit (1) as ε → 0. Various asymptotic limits, including hyperbolic limits, have been investigated in [8,9,10,11,12].The aim of this paper is the development of numerical schemes to solve the kinetic equation by methods that are uniformly stable along the transition from kinetic regime to the fluid regime. The main difficulty is due to the term 1 ε which becomes stiff when ε is close to zero (macroscopic regime). In this case, solving the kinetic equation by a standard explicit numerical scheme requires the use of a time step of the order of ε, which leads to very expensive numerical computations for small ε. To avoid this difficulty, it is necessary to use an implicit or semi-implicit time discretization for the collision part. In fact, such numerical schemes should also have a correct asymptotic behavior, namely for small parameter ε, the schemes should degenerate into a good approximation of the asymptotics (Keller-Segel model) of the kinetic equation. This property is often called "asymptotic preserving", and has been introduced in [13] for numerical schemes that are stable with respect to a small parameter ε and degenerate into a consistent numerical scheme for the limit model when ε → 0.Considering that this paper deals with asymptotic preserving scheme (AP), one also has to mention that there are different approaches to construct such schemes for kinetic models in various contexts. We mention for instance approaches based on domain decompositions, separating the macroscopic (fluid) domain from the microscopic (kinetic) one (see [14,15]). There are other kind of (AP) schemes for kinetic equations, which are ...

We consider the class of linear operator equations with operators admitting self-adjoint positive definite and m-accretive splitting (SAS). This splitting leads to an ADI-like iterative method which is equivalent to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of a minimal residual algorithm with Symmetric Gauss-Seidel and polynomial preconditioning is then applied to solve the resulting matrix operator equation. Theoretical analysis shows the convergence of the methods, and upper bounds for the decrease rate of the residual are derived. The convergence of the methods is numerically illustrated with the example of the neutron transport problem in 2-D geometry.

This paper presents an iterative method based on a self‐adjoint and m‐accretive splitting for the numerical treatment of the steady state neutron transport equation. Theoretical analysis shows that this method converges unconditionally to the unique solution of the transport equation. The convergence of the method is numerically illustrated and compared with the standard Source Iteration method and multigrid method on sample problems in slab geometry and in two dimensional space.

In this paper, we are concerned with a new stochastic system of nonlinear partial differential equations modeling the Lotka–Volterra interactions of predators and preys in the presence of prey-taxis, spatial diffusion, and noises. The spatial and temporal variations of the predator’s velocity are determined by the prey gradient. In the first part, we derive a macroscopic model from stochastic kinetic equations by using the micro–macro decomposition method. In the second part, we sketch the proof of the existence of weak martingale solutions by using a Faedo–Galerkin method. In the last part, we develop a one- and two-dimensional finite volume approximation for the stochastic kinetic and macroscopic models, respectively. Our one-dimensional space numerical scheme is uniformly stable along the transition from kinetic to macroscopic regimes. We close with various numerical tests illustrating the convergence of our numerical method and some features of our stochastic macro-scale system.