[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England, U.K., 2012, 386 pp., $90

The scalar auxiliary variable (SAV) approach for gradient flows

Cell migration is a pervasive process in many biology systems and involves protrusive forces generated by actin polymerization, myosin dependent contractile forces, and force transmission between the cell and the substrate through adhesion sites. Here we develop a computational model for cell motion that uses the phase-field method to solve for the moving boundary with physical membrane properties. It includes a reaction-diffusion model for the actin-myosin machinery and discrete adhesion sites which can be in a “gripping” or “slipping” mode and integrates the adhesion dynamics with the dynamics of the actin filaments, modeled as a viscous network. To test this model, we apply it to fish keratocytes, fast moving cells that maintain their morphology, and show that we are able to reproduce recent experimental results on actin flow and stress patterns. Furthermore, we explore the phase diagram of cell motility by varying myosin II activity and adhesion strength. Our model suggests that the pattern of the actin flow inside the cell, the cell velocity, and the cell morphology are determined by the integration of actin polymerization, myosin contraction, adhesion forces, and membrane forces.

http://absimage.aps.org/image/MAR14/MWS_MAR14-2013-000482.pdf

Coupling actin flow, adhesion, and morphology in a computational cell motility model

How to develop efficient numerical schemes while preserving energy stability at the discrete level is challenging for the three-component Cahn–Hilliard phase-field model. In this paper, we develop a set of first- and second-order temporal approximation schemes based on a novel “Invariant Energy Quadratization” approach, where all nonlinear terms are treated semi-explicitly. Consequently, the resulting numerical schemes lead to well-posed linear systems with a linear symmetric, positive definite at each time step. We prove that the developed schemes are unconditionally energy stable and present various 2D and 3D numerical simulations to demonstrate the stability and the accuracy of the schemes.

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Numerical Approximations for a three components Cahn-Hilliard phase-field Model based on the Invariant Energy Quadratization method

A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q -tensor model of liquid crystals

A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation

Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs

In this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effectiveness of the proposed methods will be demonstrated by a number of numerical examples. The numerical results also confirm that the global energy and momentum are well preserved.

Yuezheng Gong

Papers

A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q -tensor model of liquid crystals

A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation

Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs

Multi-symplectic Fourier pseudospectral method for the Kawahara equation

Structure-preserving Galerkin POD reduced-order modeling of Hamiltonian systems