A monotone conservative Eulerian–Lagrangian scheme for reaction‐diffusion‐convection equations modeling chemotaxis

Abstract

A numerical method for convection dominated diffusion problems, that exploits the use of characteristics, is derived and analyzed. It is shown to preserve positivity of solutions and be locally mass conserving. Stability, consistency and order one convergence are verified. Because of the way in which it determines characteristic pre-images of grid cells, the method can be easily implemented for 1-, 2-, or 3-dimensional problems on rectangular grids. which is to hold for x ∈ , t > 0, for a bounded domain ⊂ R d . Neumann boundary conditions were also specified: ∂u/∂ν = ∂v/∂ν = 0 (x ∈ ∂ ), where ∂/∂ν denotes the outer normal derivative along the boundary ∂ of . The variables are the cell density u(x, t) and a biochemical concentration v(x, t), which influences u(x, t) chemotactically, that is cells are compelled to move up the concentration gradient. This is reflected in the term ∇ · k 2 (u, v)∇v. When this term dominates the first on the right, the equation is said to be convection dominated, and it is this case that models some of the most interesting biological phenomena.

A mathematical model for IL6-induced differentiation of neural progenitor cells on a micropatterned polymer substrate

“…A method for discretizing the PDE u t + ∇ · (u∇ξ) = ∇ · (κ∇u) = g(u, x, y, t), (x, y) ∈ Ω, t > 0, ∂u ∂ν = 0, (x, y) ∈ ∂Ω, t > 0, utilizing a finite volume, cell-centered approach is outlined in [50]. It results in a system of the form U n+1 resultant Au = f system can be solved efficiently using the fast Fourier transform method in [51] since the matrix A has a special structure as described in [51].…”

“…This chapter includes brief descriptions of the numerical methods used for performing the simulations for the PDE system presented in Chapter 6. The solution to the PDE system was numerically approximated by using a Finite Volume approach as discussed in [50]. In the case where the diffusion coefficients were constant, the resultant system was solved using an efficient Fast Fourier Transform (FFT) method as discussed in [51]- [52].…”

“…This equation is a good candidate for a finite volume approximation due to the presence of the term ∇ · (u∇ξ(v)). A finite volume scheme has been derived in [50] for this type of equation, using properties of the characteristics to approximate the left-hand side of the PDE. The following is simply a restatement of this method's derivation, and is provided in [50].…”

“…A finite volume scheme has been derived in [50] for this type of equation, using properties of the characteristics to approximate the left-hand side of the PDE. The following is simply a restatement of this method's derivation, and is provided in [50]. These characteristics can be found for any (α, β) ∈ R i,j , yielding a characteristic preimage R i,j (t n ) that gets mapped to R i,j in ∆t under the flow of this ODE system.…”

“…This sum can be approximated using the following equations, appropriately modified for boundary cells when either no-flux or periodic boundary conditions are present. The details of the derivation are provided in [50].…”