Notes by: Andy Thaler
Yekaterina Epshteyn
Discontinuous Galerkin Finite Element Method for the
Classical Patlack-Keller-Segel Chemotaxis Model
April 14, 2011
Collaborators:
• Alina Chertock, North Carolina State University
• Ahmet Izmirlioglu, University of Pittsburgh
• David Kinderlehrer, Carnegie Mellon University
• Michal Kowalczyk, Universidad de Chile
• Alexander Kurganov, Tulane University
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Introduction
We begin with an outline of the talk, which is based on the paper (Epshteyn and Kurganov; 2008).
We will first discuss the Keller-Segel model of chemotaxis, followed by a discussion of the numerical
methods used to solve the PDEs in the model. Finally, we will present current research on the
topic.
1.1
Chemotaxis
Chemotaxis is defined as the “active orientation of cells and organisms along chemical gradients.”
There are many examples of chemotaxis in biology. In particular, animals and insects rely on an
acute sense of smell for conveying information between members of the species. Another example
can be seen in bacterial infections. When a bacterium invades the body, it may be attacked by
neutrophils (white blood cells) that move towards the bacterium as a result of chemotaxis (this
can be seen in the following video: White Blood Cell). Typically, there are two types of patterns,
namely traveling waves (e.g. periodic swarm rings or band dynamics) and aggregate formation.
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1.2
PDE-Based Chemotaxis Models
The PDEs used to model chemotaxis are usually highly nonlinear. In general, they are timedependent systems of PDEs consisting of three distinct sets of terms.
1. Reaction Terms: These terms model the interaction of different components, such as the
growth of cells, the release of chemoattractant, etc.
2. Diffusion Terms: These terms model the random motion of each component.
3. Chemotaxis Terms: These terms model the directed motion of one or more components in
response to the concentration gradient of another component (the chemoattractant).
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Keller-Segel Chemotaxis Model (Patlack 1953; K-S 1970,71)
Consider x ∈ Rn , n ≥ 1, x ∈ Ω, and t > 0. Then the Keller-Segel model for chemotaxis is

 ρt + ∇ · (χρ∇c) = ∆ρ
 ct = ∆c − c + ρ ,
subject to the Neumann boundary conditions
∇ρ · n = ∇c · n = 0,
where
1. ρ (x, t) is the cell density;
2. c (x, t) is the chemoattractant concentration;
3. χ is a chemotactic sensitivity constant;
4. Ω is a bounded domain in Rn ;
5. ∂Ω is the boundary of the domain Ω; and
6. n is a unit normal vector.
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x ∈ ∂Ω ,
(1)
The authors Herrero, Horstmann, Medina, Ni, Perthame, Velazquez, and Wang obtained the following analytical results. In 1D, solutions exist for all times. In 2D, however, global existence
depends on the “critical mass”. If the initial mass is below the threshold, the solution exists for all
times; if the initial mass lies above the threshold, the solution blows up in finite time. Capturing
this blow up is very difficult numerically.
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Numerical Methods
Finite volume methods (L. Tyson, L.G. Stern, R. LeVeque; 2000 and F.Filbet; 2006) have been
proposed to solve this system. Additionally, finite element methods have also been proposed (A.
Marrocco; 2003 and N. Saito; 2007). One of the goals is to develop a computationally efficient,
highly accurate and stable method that can capture sharp gradients. The method we choose to
use is the Discontinuous Galerkin Finite Element (DG) method. Some reasons for choosing this
method are given below:
1. flexibility to use high-order polynomial and non-polynomial basis functions;
2. ability to increase the order of approximation on each mesh element independently;
3. flexibility with the mesh choice/mesh refinement;
4. near exponential convergence on smooth solutions when an appropriate mesh is used; and
5. computational efficiency.
The first step to developing a high-order DG method is to rewrite the system (1) in an equivalent
form that is suitable for DG settings. We begin by introducing the new variables
u := cx , v := cy .
An equivalent form of the Keller-Segel system is then:

ρt + (χρu)x + (χρv)y = ∆ρ





 ct = ∆c − c + ρ


u = cx




v = cy
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(2)
with boundary conditions
∇ρ · n = ∇c · n = 0 .
This system is a system of convection-diffusion-reaction equations
kQt + F (Q)x + G (Q)y = k∆Q + R (Q) ,
(3)
where
1. Q := (ρ, c, u, v)T ;
2. F (Q) := (χρu, 0, −c, 0)T and G (Q) := (χρv, 0, 0, −c)T are the fluxes;
3. R (Q) := (0, ρ − c, −u, −v)T is the reaction term; and
4. k = 1 in the first two equations, and k = 0 in the third and fourth equations.
We note that the convective part of this system is hyperbolic. This is important! In particular,
if the system is not properly rewritten and high-order numerical methods are applied to solve the
system, the convective part may switch from being hyperbolic to being elliptic. This may lead to
major stability problems for the numerical method as propagation speeds are infinite in the elliptic
case.
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Scheme Based on DG Approximation for the Reformulated KellerSegel Model with (u, v) := ∇c
For the time discretization, we use a strong stability preserving Runge-Kutta method. For the
space discretization, we use NIPG (nonsymmetric interior penalty Galerkin), SIPG (symmetric
interior penalty Galerkin), IIPG (incomplete interior penalty Galerkin) with special treatment of
the convective terms based on central-upwind numerical fluxes, which is based on the estimation of
one-sided speeds in the x and y−direction. This scheme is very accurate and efficient when applied
to systems of conservation laws. In particular, we proved the high order error estimates in the case
when the solution remains bounded. We were also able to show that the blow up time of the actual
solution is bounded above by the blow up time of the numerical solution. In the numerical results
(see Epshteyn and Kurganov; 2008), we see that there are no negative values and that the order of
convergence of the numerical scheme is high even close to the blow up time.
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Bibliography
Y. Epshteyn and A. Kurganov. New Interior Penalty Discontinuous Galerkin Methods for the
Keller-Segel Chemotaxis Model. SIAM J. Num. Anal, 47 (2008), pp. 386-408.
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