Efficient Numerical Algorithms Based on Difference Potentials for Chemotaxis Systems in 3D

Abstract

In this work, we propose efficient and accurate numerical algorithms based on Difference Potentials Method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the use of only Cartesian meshes and employ Fast Poisson Solvers. In addition, to further enhance computational efficiency of the methods, we design a Difference-Potentialsbased domain decomposition approach which allows mesh adaptivity and easy parallelization of the algorithm in space. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the developed numerical algorithms.In this work we develop efficient and accurate numerical algorithms based on Difference Potentials Method (DPM) for the Patlak-Keller-Segel (PKS) chemotaxis model and related problems in 3D. The proposed methods handle irregular geometry with the use of only Cartesian meshes, employ Fast Poisson Solvers, allow easy parallelization and adaptivity in space.Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, for instance a chemical one. Chemotaxis is an essential process in many medical and biological applications, including cell aggregation and pattern formation mechanisms and tumor growth. Modeling of chemotaxis dates back to the pioneering work by Patlak, Keller and Segel [28,29,38]. The PKS chemotaxis model consists of coupled convection-diffusion and reaction-diffusion partial differential equations:subject to zero Neumann boundary conditions and the initial conditions on the cell density ρ(x, y, z, t) and on the chemoattractant concentration c(x, y, z, t). In the system (1), the coefficient χ is a chemotactic sensitivity constant, γ ρ and γ c are the reaction coefficients. The parameter α is equal to either 1 or 0, which corresponds to the "parabolic-parabolic" or reduced "parabolic-elliptic" coupling, respectively. In this work, we will focus on the PKS chemotaxis model (1) with α = γ ρ = γ c = 1, but the developed methods are not restricted by this assumption and can be easily extended to more general chemotaxis systems and related models.In the past several years, chemotaxis models have been extensively studied (see, e.g., [19,20,21,22, 39] and references below). A known property of many chemotaxis systems is their ability to represent a concentration phenomenon that is mathematically described by fast growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular behavior. This blowup represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems, see, e.g., [1,5,6,7,11,12,36,40].Capturing blowing up or spiky solutions is a challenging task numerically, but at the same time design of robust numerical algorithms is crucial for the modeling and analysis of chemotaxis mechanisms. Let us briefly review some of the recent numerical methods that have been proposed for the "parabolic-parabolic" cou...

Difference Potentials Method for Models with Dynamic Boundary Conditions and Bulk-Surface Problems

“…Proof. The proof follows the lines of the proof in [18,33] and we will present it below for reader's convenience. First, define the grid function:…”

“…The current work is a continuation of the recent work in [2,3,18,23]. For the time being, we will consider the model with dynamic boundary conditions and the bulk-surface problem in a spherical domain, but the proposed methods can be extended to domains with more general geometry in 3D (and the main ideas of the algorithms will stay the same, see Remark 5 below).…”

“…For the time being, we will consider the model with dynamic boundary conditions and the bulk-surface problem in a spherical domain, but the proposed methods can be extended to domains with more general geometry in 3D (and the main ideas of the algorithms will stay the same, see Remark 5 below). We employ a finite-difference scheme for the underlying space discretization of the models in the bulk (1) or ( 4), combined with the idea of Difference Potentials Method (DPM) ( [33] and very recent work [2,3,17,18,32], etc.) that provides flexibility to handle irregular domains and nontrivial boundary conditions (including, but not limited to, dynamic boundary conditions like (2), or surface equations like (6)) accurately and efficiently.…”

“…Proof. The proof follows the lines of the proof in [18,33], and we will present it below for reader's convenience. If the density u i+1 γ e x on γ ex to the difference equation ( 10) is given, then such discrete system will admit a unique solution u i+1 j,k,l defined on a set N + .…”

“…Next, observe that the BEP (15) and the reduced BEP (17) can be reformulated using grid function Φ i+1 in (18) as follows:…”