Publications 2019

An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions
A. Chertock, A. Kurganov, M. Lukacova-Medvidova, S. Nur Oezcan
Kinetic and Related Models 12 (1), 195–216 (2019);
URL: http://aimsciences.org//article/doi/10.3934/krm.2019009
doi:10.3934/krm.2019009

↓ Show Abstract↑ Hide Abstract In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart-the Patlak-Keller-Segel system-in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.

Energy-stable linear schemes for polymer-solvent phase field models
P. Strasser, G. Tierra, B. Dünweg, M. Lukacova-Medvidova
Comp. Math. Appl. 77 (1), 125-143 (2019);
URL: https://www.sciencedirect.com/science/article/pii/S0898122118305303?via%3Dihub
doi:https://doi.org/10.1016/j.camwa.2018.09.018

↓ Show Abstract↑ Hide Abstract We present new linear energy-stable numerical schemes for numerical simulation of complex polymer–solvent mixtures. The mathematical model proposed by Zhou et al. (2006) consists of the Cahn–Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymer–solvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties.