A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses

A 1D1V hybrid Vlasov-fluid model was developed for this study to elucidate ionization oscillations of Hall thrusters (HTs). The Vlasov equation for ions velocity distribution function (IVDF) with ionization source term is solved using a constrained interpolation profile conservative semi-Lagrangian (CIP-CSL) method. The fourth-order weighted essentially non-oscillatory (4th WENO) limiter is applied to the first derivative term to minimize numerical oscillation in the discharge oscillation analyses. The fourth-order spatial accuracy is verified through a 1D scalar test case. Nonoscillatory and high-resolution features of the Vlasov model are confirmed by simulating the test cases of the Vlasov–Poisson (VP) system and by comparing the results with a particle-in-cell (PIC) method. A 1D1V Hall thruster simulation is performed through the hybrid Vlasov-fluid model. The ionization oscillation is analysed. The macroscopic plasma properties are compared with those obtained from a hybrid PIC method. The comparison indicates that the hybrid Vlasov-fluid model yields noiseless results and that the steady-state waveform is calculable in a short time period.

Past and Present Trends in the Development of the Pattern-Formation Theory: Domain Walls and Quasicrystals

A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying semi-infinite regions, and (ii) two- and three-dimensional (2D and 3D) quasiperiodic (QP) patterns, which are built as a superposition of plane–wave modes with incommensurate spatial periodicities. These topics are selected for the present review, dedicated to the 70th birthday of Professor Michael I. Tribelsky, due to the impact made on them by papers of Prof. Tribelsky and his coauthors. Although some findings revealed in those works may now seem “old”, they keep their significance as fundamentally important results in the theory of nonlinear DW and QP patterns. Adding to the findings revealed in the original papers by M.I. Tribelsky et al., the present review also reports several new analytical results, obtained as exact solutions to systems of coupled real Ginzburg–Landau (GL) equations. These are a new solution for symmetric DWs in the bimodal system including linear mixing between its components; a solution for a strongly asymmetric DWs in the case when the diffusion (second-derivative) term is present only in one GL equation; a solution for a system of three real GL equations, for the symmetric DW with a trapped bright soliton in the third component; and an exact solution for DWs between counter-propagating waves governed by the GL equations with group-velocity terms. The significance of the “old” and new results, collected in this review, is enhanced by the fact that the systems of coupled equations for two- and multicomponent order parameters, addressed in this review, apply equally well to modeling thermal convection, multimode light propagation in nonlinear optics, and binary Bose–Einstein condensates.

Radius ratio dependency of the instability of fully compressible convection in rapidly rotating spherical shells

Based on the fully compressible Navier–Stokes equations, the linear stability of thermal convection in rapidly rotating spherical shells of various radius ratios $\eta$ is studied for a wide range of Taylor number $Ta$ , Prandtl number $Pr$ and the number of density scale height $N_\rho$ . Besides the classical inertial mode and columnar mode, which are widely studied by the Boussinesq approximation and anelastic approximation, the quasi-geostrophic compressible mode is also identified in a wide range of $N_\rho$ and $Pr$ for all $\eta$ considered, and this mode mainly occurs in the convection with relatively small $Pr$ and large $N_\rho$ . The instability processes are classified into five categories. In general, for the specified wavenumber $m$ , the parameter space ( $Pr, N_\rho$ ) of the fifth category, in which the base state loses stability via the quasi-geostrophic compressible mode and remains unstable, shrinks as $\eta$ increases. The asymptotic scaling behaviours of the critical Rayleigh numbers $Ra_c$ and corresponding wavenumbers $m_c$ to $Ta$ are found at different $\eta$ for the same instability mode. As $\eta$ increases, the flow stability is strengthened. Furthermore, the linearized perturbation equations and Reynolds–Orr equation are employed to quantitatively analyse the mechanical mechanisms and flow instability mechanisms of different modes. In the quasi-geostrophic compressible mode, the time-derivative term of disturbance density in the continuity equation and the diffusion term of disturbance temperature in the energy equation are found to be critical, while in the columnar and inertial modes, they can generally be ignored. Because the time-derivative term of the disturbance density in the continuity equation cannot be ignored, the anelastic approximation fails to capture the instability mode in the small- $Pr$ and large- $N_\rho$ system, where convection onset is dominated by the quasi-geostrophic compressible mode. However, all the modes are primarily governed by the balance between the Coriolis force and the pressure gradient, based on the momentum equation. Physically, the most important difference between the quasi-geostrophic compressible mode and the columnar mode is the role played by the disturbance pressure. The disturbance pressure performs negative work for the former mode, which appears to stabilize the flow, while it destabilizes the flow for the latter mode. As $\eta$ increases, in the former mode the relative work performed by the disturbance pressure increases and in the latter mode decreases.

Families of Solutions of Multitemporal Nonlinear Schrödinger PDE

The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of the envelope of a spatially and temporarily oscillating wave packet in space and multitime (an equation which governs the dynamics of solitons through meta-materials). Now we exploit some hypotheses in order to find important explicit families of exact solutions in all dimensions for the multitime nonlinear Schrödinger PDE with a multitemporal directional derivative term. Using quite effective methods, we discovered families of ODEs and PDEs whose solutions generate solutions of multitime nonlinear Schrödinger PDE. Each new construction involves a relatively small amount of intermediate calculations.

Attempt to Describe Phase Slips by Means of an Adiabatic Approximation

As a plausibility test for the feasibility of extension of the quasiclassical Keldysh–Usadel technique to slowly varying situations, we assess the influence of the time-derivative term in the time-dependent Ginzburg–Landau equation. We consider cases in which the superconducting state in a nanowire varies slowly, either because the voltage applied on it is small, or because most of phase drift takes place next to the boundaries. An approximation without this time derivative can describe the superconducting state away from phase slips, but is unable to predict the value or the existence of a critical voltage at which evolution becomes non-stationary.

Holographic phonons by gauge-axion coupling

In this paper, we show that a simple generalization of the holographic axion model can realize spontaneous breaking of translational symmetry by considering a special gauge-axion higher derivative term. The finite real part and imaginary part of the stress tensor imply that the dual boundary system is a viscoelastic solid. By calculating quasi-normal modes and making a comparison with predictions from the elasticity theory, we verify the existence of phonons and pseudo-phonons, where the latter is realized by introducing a weak explicit breaking of translational symmetry, in the transverse channel. Finally, we discuss how the phonon dynamics affects the charge transport.

Large charge sector of 3d parity-violating CFTs

Certain CFTs with a global U(1) symmetry become superfluids when coupled to a chemical potential. When this happens, a Goldstone effective field theory controls the spectrum and correlators of the lightest large charge operators. We show that in 3d, this EFT contains a single parity-violating 1-derivative term with quantized coefficient. This term forces the superfluid ground state to have vortices on the sphere, leading to a spectrum of large charge operators that is remarkably richer than in parity-invariant CFTs. We test our predictions in a weakly coupled Chern-Simons matter theory.

Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System

The present study uses linear quadratic regulator (LQR) theory to control a vibratory system modeled by a fractional-order differential equation. First, as an example of such a vibratory system, a viscoelastically damped structure is selected. Second, a fractional-order LQR is designed for a system in which fractional-order differential terms are contained in the equation of motion. An iteration-based method for solving the algebraic Riccati equation is proposed in order to obtain the feedback gains for the fractional-order LQR. Third, a fractional-order state observer is constructed in order to estimate the states originating from the fractional-order derivative term. Fourth, numerical simulations are presented using a numerical calculation method corresponding to a fractional-order state equation. Finally, the numerical simulation results demonstrate that the fractional-order LQR control can suppress vibrations occurring in the vibratory system with viscoelastic damping.

Comparison of different time discretization schemes for solving the Allen–Cahn equation

This article aims to solve the nonlinear Allen–Cahn equation numerically. The diagonally implicit fractional-step θ-(DIFST) scheme is used for the discretization of the time derivative term while the space derivative is discretized by the conforming finite element method. The computational efficiency of the DIFST scheme in terms of CPU time and temporal error estimation is computed and compared with other time discretization schemes. Several test problems are presented to show the effectiveness of the DIFST scheme.