Michael Fischer, Laura Kanzler, Christian Schmeiser
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引用次数: 0
Abstract
SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 1-18, February 2024. Abstract. Repulsion between individuals within a finite radius is encountered in numerous applications, including cell exclusion, i.e., avoidance of overlapping cells, bird flocks, or microscopic pedestrian models. We define such individual-based particle dynamics in one spatial dimension with minimal assumptions of the repulsion force [math] as well as their external velocity [math] and prove their characteristic properties. Moreover, we are able to perform a rigorous limit from the microscopic to the macroscopic scale, where we could recover the finite interaction radius as a density threshold. Specific choices for the repulsion force [math] lead to well-known nonlinear diffusion equations on the macroscopic scale, as, e.g., the porous medium equation. At both scaling levels, numerical simulations are presented and compared to underline the analytical results.
期刊介绍：
SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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