Dynamics of Structured Systems

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Discrete momentum maps for lattice EPDiff

We focus on the spatial discretization produced by the Variational Particle-Mesh (VPM) method for a prototype fluid equation the known as the EPDiff equation}, which is short for Euler-Poincar\'e equation associated with the diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold $\Omega$). The EPDiff equation admits measure valued solutions, whose dynamics are determined by the momentum maps for the left and right actions of the diffeomorphisms on embedded subspaces of $\mathbb{R}^d$. The discrete VPM analogs of those dynamics are studied here. Our main results are: (i) a variational formulation for the VPM method, expressed in terms of a constrained variational principle principle for the Lagrangian particles, whose velocities are restricted to a distribution D_{\VPM} which is a finite-dimensional subspace of the Lie algebra of vector fields on $\Omega$; (ii) a corresponding constrained variational principle on the fixed Eulerian grid which gives a discrete version of the Euler-Poincar\'e equation; and (iii) discrete versions of the momentum maps for the left and right actions of diffeomorphisms on the space of solutions

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

The ADI-FDTD Method for High Accuracy Electrophysics Applications

The Finite-Difference Time-Domain (FDTD) is a dependable method to simulate a wide range of problems from acoustics, to electromagnetics, and to photonics, amongst others. The execution time of an FDTD simulation is inversely proportional to the time-step size. Since the FDTD method is explicit, its time-step size is limited by the well-known Courant-Friedrich-Levy (CFL) stability limit. The CFL stability limit can render the simulation inefficient for very fine structures. The Alternating Direction Implicit FDTD (ADI-FDTD) method has been introduced as an unconditionally stable implicit method. Numerous works have shown that the ADI-FDTD method is stable even when the CFL stability limit is exceeded. Therefore, the ADI-FDTD method can be considered an efficient method for special classes of problems with very fine structures or high gradient fields. Whenever the ADI-FDTD method is used to simulate open-region radiation or scattering problems, the implementation of a mesh-truncation scheme or absorbing boundary condition becomes an integral part of the simulation. These truncation techniques represent, in essence, differential operators that are discretized using a distinct differencing scheme which can potentially affect the stability of the scheme used for the interior region. In this work, we show that the ADI-FDTD method can be rendered unstable when higher-order mesh truncation techniques such as Higdon's Absorbing Boundary Condition (ABC) or Complementary Derivatives Method (COM) are used. When having large field gradients within a limited volume, a non-uniform grid can reduce the computational domain and, therefore, it decreases the computational cost of the FDTD method. However, for high-accuracy problems, different grid sizes increase the truncation error at the boundary of domains having different grid sizes. To address this problem, we introduce the Complementary Derivatives Method (CDM), a second-order accurate interpolation scheme. The CDM theory is discussed and applied to numerical examples employing the FDTD and ADI-FDTD methods

Topics in Magnetohydrodynamics

To understand plasma physics intuitively one need to master the MHD behaviors. As sciences advance, gap between published textbooks and cutting-edge researches gradually develops. Connection from textbook knowledge to up-to-dated research results can often be tough. Review articles can help. This book contains eight topical review papers on MHD. For magnetically confined fusion one can find toroidal MHD theory for tokamaks, magnetic relaxation process in spheromaks, and the formation and stability of field-reversed configuration. In space plasma physics one can get solar spicules and X-ray jets physics, as well as general sub-fluid theory. For numerical methods one can find the implicit numerical methods for resistive MHD and the boundary control formalism. For low temperature plasma physics one can read theory for Newtonian and non-Newtonian fluids etc