Fractional step methods applied to a chemotaxis model

Abstract

A fractional step numerical method is developed for the nonlinear partial differential equations arising in chemotaxis models, which include density-dependent diffusion terms for chemotaxis, as well as reaction and Fickian diffusion terms. We take the novel approach of viewing the chemotaxis term as an advection term which is possible in the context of fractional steps. This method is applied to pattern formation problems in bacterial growth and shown to give good results. High-resolution methods capable of capturing steep gradients (from CLAWPACK) are used for the advection step, while the A-stable and L-stable TR-BDF2 method is used for the diffusion step. A numerical instability that is seen with other diffusion methods is analyzed and eliminated.

New Interior Penalty Discontinuous Galerkin Methods for the Keller–Segel Chemotaxis Model

“…A finite-volume, [21], and a finite-element, [34], methods have been proposed for a simpler version of the Keller-Segel model, ρ t + ∇ · (χρ∇c) = ∆ρ, ∆c − c + ρ = 0, in which the equation for concentration c has been replaced by an elliptic equation using an assumption that the chemoattractant concentration c changes over much smaller time scales than the density ρ. A fractional step numerical method for a fully time-dependent chemotaxis system from [41] has been proposed in [42]. However, the operator splitting approach may not be applicable when a convective part of the chemotaxis system is not hyperbolic, which is a generic situation for the original Keller-Segel model as it was shown in [12], where the finitevolume Godunov-type central-upwind scheme was derived for (1.1) and extended to some other chemotaxis and haptotaxis models.…”

Numerical option pricing without oscillations using flux limiters

“…From Moreover, it is clear that F GF χ and GF G χ yield second order convergence for the European call option while F GF 1 and GF G 1 in some cases fail to do that as predicted by theory in (13). Also the accuracy in u is deteriorating for increasing s with F GF 1 and GF G 1 in Figure 2.…”

“…The approximation of u n is of formal second order in ∆t in (13) and the same factor rsχ is multiplying both u s and u ss as assumed in the derivation of the limiters in [7,Ch. 6].…”

“…This approximation is of second order in the approximation of the s-derivatives in (13). The order of accuracy is lowered to one at extrema of the solution by the limiter [7,Ch.…”

Accurate discretization of advection-diffusion equations

“…There are two major challenges: stability, which can be improved using a range of implicit methods, and accuracy, which is a delicate issue, requiring the "best possible" form of spatial discretization. Regarding the issue of stability, many schemes are in use, such as the Crank-Nicholson/ADI and fractional step methods [12,13], and the Lax-Wendroff method [14]. The issue of accuracy has received somewhat less attention with two spatial discretization schemes (and their immediate variants) commonly in use: these are the simple…”