168 research outputs found
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
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Small Collaboration: Advanced Numerical Methods for Nonlinear Hyperbolic Balance Laws and Their Applications (hybrid meeting)
This small collaborative workshop brought together
experts from the Sino-German project working in the field of advanced numerical methods for
hyperbolic balance laws. These are particularly important for compressible fluid flows and related systems of equations. The investigated numerical methods were finite volume/finite difference, discontinuous Galerkin methods, and kinetic-type schemes. We have discussed challenging open mathematical research problems in this field, such as multidimensional shock waves, interfaces with different phases or efficient and problem suited adaptive algorithms. Consequently, our main objective was to discuss novel high-order accurate schemes that reliably approximate underlying physical models and preserve important physically relevant properties. Theoretical questions concerning the
convergence of numerical methods and proper solution concepts were addressed as well
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
A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System
We introduce an efficient stochastic interacting particle-field (SIPF)
algorithm with no history dependence for computing aggregation patterns and
near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis
system in three space dimensions (3D). The KS solutions are approximated as
empirical measures of particles coupled with a smoother field (concentration of
chemo-attractant) variable computed by the spectral method. Instead of using
heat kernels causing history dependence and high memory cost, we leverage the
implicit Euler discretization to derive a one-step recursion in time for
stochastic particle positions and the field variable based on the explicit
Green's function of an elliptic operator of the form Laplacian minus a positive
constant. In numerical experiments, we observe that the resulting SIPF
algorithm is convergent and self-adaptive to the high gradient part of
solutions. Despite the lack of analytical knowledge (e.g. a self-similar
ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study
the emergence of finite time blowup in 3D by only dozens of Fourier modes and
through varying the amount of initial mass and tracking the evolution of the
field variable. Notably, the algorithm can handle at ease multi-modal initial
data and the subsequent complex evolution involving the merging of particle
clusters and formation of a finite time singularity
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