Mathematics
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Item The emergence of lines of hierarchy in collective motion of biological systems(Institute of Physics, 2023-06-29) Greene, James M.; Tadmor, Eitan; Zhong, MingThe emergence of large-scale structures in biological systems, and in particular the formation of lines of hierarchy, is observed at many scales, from collections of cells to groups of insects to herds of animals. Motivated by phenomena in chemotaxis and phototaxis, we present a new class of alignment models that exhibit alignment into lines. The spontaneous formation of such ‘fingers’ can be interpreted as the emergence of leaders and followers in a system of identically interacting agents. Various numerical examples are provided, which demonstrate emergent behaviors similar to the ‘fingering’ phenomenon observed in some phototaxis and chemotaxis experiments; this phenomenon is generally known to be a challenging pattern for existing models to capture. A novel protocol for pairwise interactions provides a fundamental alignment mechanism by which agents may form lines of hierarchy across a wide range of biological systems.Item Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: high-order semi-discrete schemes.(Copyright: Society for Industrial and Applied Mathematics, 2006) Balbas, Jorge; Tadmor, EitanWe present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balb´as, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261–285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241–282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio–Wu shock-tube problems and the two-dimensional Kelvin–Helmholtz instability, Orszag–Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the ∇ · B = 0-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.Item High order time discretization methods with the strong stability property(Copyright: Society for Industrial and Applied Mathematics, 2001) Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, EitanIn this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations.Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations.The new developments in this paper include the construction of optimal explicit SSP linear Runge–Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge–Kutta and multistep methods.