A significant step towards the comprehensive solution of the important new fundamental control problems of high-temperature plasma of toroidal configuration is made by utilizing heterogeneous mini-supercomputers of NIISI RAS (Scientific Research Institute of System Development, Russian Academy of Science). The basic components of hardware and software modeling system and automatic real-time plasma feedback control are developed and implemented. The system is named HASP CS (Hardware-Software Plasma Control System). It is based on a fundamentally new, more accurate and reliable control algorithm using a solution with a given accuracy of a number of inverse diagnostic problems by $ϵ$-nets. The possibility of an application of the proposed technique during discharge time to provide feedback control of the boundary and internal plasma parameters is demonstrated. It is shown that the complex allows investigating the processes in currently operated and designed fusion devices and solving problems of fusion application for energy production. The recommendations to the engineering implementation of plasma control systems are made. The directions for further research are formulated.

We introduce a variational method for demodulating phase maps from fringe patterns. This new method is based on the mean curvature of the level sets ofthe phase surface that is used for regularization. The performance of the method is illustrated with both synthetic and real data.

The minimum action method (MAM) is to calculate the most probable transition path in randomly perturbed stochastic dynamics, based on the idea of action minimization in the path space. The accuracy of the numerical path between different metastable states usually suffers from the "clustering problem" near fixed points. The adaptive minimum action method (aMAM) solves this problem by relocating image points equally along arc-length with the help of moving mesh strategy. However, when the time interval is large, the images on the path may still be locally trapped around the transition state in a tangle, due to the singularity of the relationship between arc-length and time at the transition state. Additionally, in most non-gradient dynamics, the tangent direction of the path is not continuous at the transition state so that a geometric corner forms, which brings extra challenges for the aMAM. In this note, we improve the aMAM by proposing a better monitor function that does not contain the numerical approximation of derivatives, and taking use of a generalized scheme of the Euler-Lagrange equation to solve the minimization problem, so that both the path-tangling problem and the non-smoothness in parametrizing the curve do not exist. To further improve the accuracy, we apply the Weighted Essentially non-oscillatory (WENO) method for the interpolation to achieve better performance. Numerical examples are presented to demonstrate the advantages of our new method.

Many physical problems involve unbounded domains where the physical quantities vanish at infinities. Numerically, this has been handled using different techniques such as domain truncation, approximations using infinitely extended and vanishing basis sets, and mapping bounded basis sets using some coordinate transformations. Each technique has its own advantages and disadvantages. Yet, approximating simultaneously and efficiently a wide range of decaying rates has persisted as major challenge. Also, coordinate transformation, if not carefully implemented, can result in non-orthogonal mapped basis sets. In this work, we revisited this issue with an emphasize on designing appropriate transformations using sine series as basis set. The transformations maintain both the orthogonality and the efficiency. Furthermore, using simple basis set (sine function) help avoid the expensive numerical integrations. In the calculations, four types of physically recurring decaying behaviors are considered, which are: non-oscillating and oscillating exponential decays, and non-oscillating and oscillating algebraic decays. The results and the analyses show that properly designed high-order mapped basis sets can be efficient tools to handle challenging physical problems on unbounded domains. Decay rate ranges as large of 6 orders of magnitudes can be approximated efficiently and concurrently.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods. The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

Lattice Boltzmann model (LBM) in conjunction with an accurate Large Eddy Simulation (LES) technology was proposed to simulate various vortical structures and their evolutions in open pump intakes. The strain rate tensor in the LES model is locally calculated by means of non-equilibrium moments based on Chapman-Enskog expansion, and bounce-back scheme was used for non-slip condition on solid walls and reflection scheme for free surface. The presented model was applied to investigate free-surface and wall-attached vortices for different water levels and flow rate. The vortex position, shapes and vorticities were predicted successfully under three flowing cases (i.e. critical water level (CWL), lower water level, lower flow rate), and the numerical velocity and streamline distribution were analyzed systematically. For CWL based on Froude number considering open channel flows, the shape and the location of various dynamic vortices were captured. Compare to the experimental results of CWL, more vortices were predicted for lower water level, and less vortices were observed for lower flow rate. The predicted velocities and vortex locations are in good agreement with the experimental of a small physical model. The comparisons demonstrated the feasibility and stability of above-mentioned model and numerical method in predicting vortex flows inside open pump intakes.

In this paper, we will develop a first order and a second order convex splitting, and a first order linear energy stable fully discrete local discontinuous Galerkin (LDG) methods for the modified phase field crystal (MPFC) equation. In which, the first order linear scheme is based on the invariant energy quadratization approach. The MPFC equation is a damped wave equation, and to preserve an energy stability, it is necessary to introduce a pseudo energy, which all increase the difficulty of constructing numerical methods comparing with the phase field crystal (PFC) equation. Due to the severe time step restriction of explicit time marching methods, we introduce the first order and second order semi-implicit schemes, which are proved to be unconditionally energy stable. In order to improve the temporal accuracy, the semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting scheme is employed. Numerical simulations of the MPFC equation always need long time to reach steady state, and then adaptive time-stepping method is necessary and of paramount importance. The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver. Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods, and the effectiveness of the adaptive time-stepping strategy.

The linear hyperbolic equation is of great interest in many branches of physics and industry. In this paper, we use the weak Galerkin method to solve the linear hyperbolic equation. Since the weak Galerkin finite element space consists of discontinuous polynomials, the discontinuous feature of the equation can be maintained. The optimal error estimates are proved. Some numerical experiments are provided to verify the efficiency of the method.

A compressible and multiphase flows solver has been developed for the study of liquid/gas flows involving shock waves and strong expansion waves leading to cavitation. This solver has a structure similar to those of the one-fluid Euler solvers, differing from them by the presence of a void ratio transport-equation. The model and the system of equations to be simulated are presented. Results are displayed for shock and expansion tube problems, shock-bubble interaction and underwater explosion. Close agreement with reference solutions, obtained from explicit finite volume approaches, is demonstrated. Different numerical methods are additionally displayed to provide comparable and improved computational efficiency to the model and the system of equations. The overall procedure is therefore very well suited for use in general two-phase fluid flow simulations.

This paper develops a second-order multiscale asymptotic analysis and numerical algorithms for predicting heat transfer performance of porous materials with quasi-periodic structures. In these porous materials, they have periodic configurations and associated coefficients are dependent on the macro-location. Also, radiation effect at microscale has an important influence on the macroscopic temperature fields, which is our particular interest in this study. The characteristic of the coupled multiscale model between macroscopic scale and microscopic scale owing to quasi-periodic structures is given at first. Then, the second-order multiscale formulas for solving temperature fields of the nonlinear problems are constructed, and associated explicit convergence rates are obtained on some regularity hypothesis. Finally, the corresponding finite element algorithms based on multiscale methods are brought forward and some numerical results are given in detail. Numerical examples including different coefficients are given to illustrate the efficiency and stability of the computational strategy. They show that the expansions to the second terms are necessary to obtain the thermal behavior precisely, and the local and global oscillations of the temperature fields are dependent on the microscopic and macroscopic part of the coefficients respectively.

Red blood cells can recover their resting shape after having been deformed by shear flow. Their rims are always formed by the same part of the membranes, and the cells are said to have shape memory. Modeled as two-dimensional elastic capsules, their recovery motion and shape memory is studied, mainly focused on the effect of the spontaneous shape. The fluid-structure interaction is modeled using immersed boundary method. Based on the simulations, the resting shapes of capsules are obtained and the area ratio of spontaneous shape is found to play an important role. After remove of shear flow, all capsules can recover their resting shapes, while only capsules with noncircular spontaneous shapes present shape memory. As the spontaneous shape approaches a circle but still noncircular, the capsule spends more time on recovery process. We consider how these capsules deform depending on the membrane bending energy, and find that the relaxation speed is positive correlated to the range of values of dimensionless bending energy. These results may help to identify different spontaneous shapes for capsules especially RBCs through future experiments.

This work addresses the nonlinear dynamic behavior of different electrostatic micro-tweezers, a micro electric actuator. This actuator, a cantilever beam electrostatic micro-tweezers, has been extensively used in micro-electro-mechanical systems (MEMS). The importance of micro electric actuators manufactured is higher than the other part of MEMS since it is the power source of the entire micro-electro-mechanical systems. In actual operation, the instability and bad dynamic characteristics of the electric actuators will cause larger displacement mobility error, such as transport behavior and response procedures failure, etc., and even damage the micro-electro-mechanical systems. To improve the actuator dynamic displacement accuracy, the dynamic behavior in the electric actuator system must be studied, especially for nonlinear dynamic behavior of system. In this work, the differential quadrature method (DQM) was employed to solve the problem of nonlinearity in the equation of motion. The results reveal that the proposed DQM model can be used to simulate the nonlinear behavior of the micro-tweezers efficiently. Micro-tweezers of various shapes were studied to examine the feasibility of applying the DQM in analyzing their nonlinear responses. The simulated results agree very closely with the calculated and experimental data in the literature.

In a recent paper (Du and Ekaterinaris, 2016) optimization of dissipation and dispersion errors was investigated. A Diagonally Implicit Runge-Kutta (DIRK) scheme was developed by using the relative stability concept, i.e. the ratio of absolute numerical stability function to analytical one. They indicated that their new scheme has many similarities to one of the optimized Strong Stability Preserving (SSP) schemes. They concluded that, for steady state simulations, time integration schemes should have high dissipation and low dispersion. In this note, dissipation and dispersion errors for DIRK schemes are studied further. It is shown that relative stability is not an appropriate criterion for numerical stability analyses. Moreover, within absolute stability analysis, it is shown that there are two important concerns, accuracy and stability limits. It is proved that both A-stability and SSP properties aim at minimizing the dissipation and dispersion errors. While A-stability property attempts to increase the stability limit for large time step sizes and by bounding the error propagations via minimizing the numerical dispersion relation, SSP optimized method aims at increasing the accuracy limits by minimizing the difference between analytical and numerical dispersion relations. Hence, it can be concluded that A-stability property is necessary for calculations under large time-step sizes and, more specifically, for calculation of high diffusion terms. Furthermore, it is shown that the oscillatory behavior, reported by Du and Ekaterinaris (2016), is due to Newton method and the tolerances they set and it is not related to the employed temporal schemes.

In this paper, we discuss a gradient-enhanced $ℓ_1$ approach for the recovery of sparse Fourier expansions. By $gradient$-$enhanced$ approaches we mean that the directional derivatives along given vectors are utilized to improve the sparse approximations. We first consider the case where both the function values and the directional derivatives at sampling points are known. We show that, under some mild conditions, the inclusion of the derivatives information can indeed decrease the coherence of measurement matrix, and thus leads to the improved the sparse recovery conditions of the $ℓ_1$ minimization. We also consider the case where either the function values or the directional derivatives are known at the sampling points, in which we present a sufficient condition under which the measurement matrix satisfies RIP, provided that the samples are distributed according to the uniform measure. This result shows that the derivatives information plays a similar role as that of the function values. Several numerical examples are presented to support the theoretical statements. Potential applications to function (Hermite-type) interpolations and uncertainty quantification are also discussed.