34 research outputs found
A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model
We provide a posteriori error estimates for a discontinuous Galerkin scheme
for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The
estimates are conditional, in the sense that an a posteriori computable
quantity needs to be small enough - which can be ensured by mesh refinement -
and optimal in the sense that the error estimator decays with the same order as
the error under mesh refinement. A specific feature of our error estimator is
that it can be used to prove existence of a weak solution up to a certain time
based on numerical results.Comment: 31 pages, 1 figure, 5 table
A study of blow-ups in the Keller-Segel model of chemotaxis
We study the Keller-Segel model of chemotaxis and develop a composite
particle-grid numerical method with adaptive time stepping which allows us to
accurately resolve singular solutions. The numerical findings (in two
dimensions) are then compared with analytical predictions regarding formation
and interaction of singularities obtained via analysis of the stochastic
differential equations associated with the Keller-Segel model
An unconditionally energy stable and positive upwind DG scheme for the Keller-Segel model
The well-suited discretization of the Keller-Segel equations for chemotaxis
has become a very challenging problem due to the convective nature inherent to
them. This paper aims to introduce a new upwind, mass-conservative, positive
and energy-dissipative discontinuous Galerkin scheme for the Keller-Segel
model. This approach is based on the gradient-flow structure of the equations.
In addition, we show some numerical experiments in accordance with the
aforementioned properties of the discretization. The numerical results obtained
emphasize the really good behaviour of the approximation in the case of
chemotactic collapse, where very steep gradients appear.Comment: 24 pages, 17 figures, 4 table
A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model
We study a finite volume scheme approximating a parabolic-elliptic
Keller-Segel system with power law diffusion with exponent
and periodic boundary conditions. We derive conditional a posteriori bounds for
the error measured in the norm for the
chemoattractant and by a quasi-norm-like quantity for the density. These
results are based on stability estimates and suitable conforming
reconstructions of the numerical solution. We perform numerical experiments
showing that our error bounds are linear in mesh width and elucidating the
behaviour of the error estimator under changes of .Comment: 26 pages, 2 figures, 3 table
Bound-preserving finite element approximations of the Keller-Segel equations
This paper aims to develop numerical approximations of the Keller--Segel
equations that mimic at the discrete level the lower bounds and the energy law
of the continuous problem. We solve these equations for two unknowns: the
organism (or cell) density, which is a positive variable, and the
chemoattractant density, which is a nonnegative variable. We propose two
algorithms, which combine a stabilized finite element method and a
semi-implicit time integration. The stabilization consists of a nonlinear
artificial diffusion that employs a graph-Laplacian operator and a shock
detector that localizes local extrema. As a result, both algorithms turn out to
be nonlinear.Both algorithms can generate cell and chemoattractant numerical
densities fulfilling lower bounds. However, the first algorithm requires a
suitable constraint between the space and time discrete parameters, whereas the
second one does not. We design the latter to attain a discrete energy law on
acute meshes. We report some numerical experiments to validate the theoretical
results on blowup and non-blowup phenomena. In the blowup setting, we identify
a \textit{locking} phenomenon that relates the -norm to the
-norm limiting the growth of the singularity when supported on a
macroelement.Comment: 27 pages, 22 figure
Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations
Keller-Segel systems are a set of nonlinear partial differential equations
used to model chemotaxis in biology. In this paper, we propose two alternating
direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly
with minimal computational cost, while preserving positivity, energy
dissipation law and mass conservation. One scheme unconditionally preserves
positivity, while the other does so conditionally. Both schemes achieve
second-order accuracy in space, with the former being first-order accuracy in
time and the latter second-order accuracy in time. Besides, the former scheme
preserves the energy dissipation law asymptotically. We validate these results
through numerical experiments, and also compare the efficiency of our schemes
with the standard five-point scheme, demonstrating that our approaches
effectively reduce computational costs.Comment: 29 page