Energy-conserving Scheme Research Articles

Theoretical analysis of a conservative finite-difference scheme to solve a Riesz space-fractional Gross–Pitaevskii system

In this work, we propose a fractional extension of the multi-dimensional Gross–Pitaevskii system that describes a two-component Bose–Einstein condensate with an internal atomic Josephson junction. The fractional problem is governed by two parabolic partial differential equations that consider fractional spatial derivatives of the Riesz type along with coupling terms. Initial and homogeneous Dirichlet boundary conditions are imposed on a bounded interval of a closed and bounded domain. We show that the problem can be expressed in variational form and propose a Hamiltonian function associated to the system. We prove that the total energy of the system is constant, whence the need to provide energy-conserving schemes to solve the system is pragmatically justified. Motivated by these facts, we design a finite-difference discretization of the continuous model based on the use of fractional-order centered differences. The discrete scheme has also a variational structure, and we propose a discrete form of the Hamiltonian function. As the continuous counterpart, we prove rigorously that the discrete total energy is conserved at each temporal step. The scheme is a second-order consistent discretization of the continuous model. Moreover, we prove the stability and quadratic convergence of the numerical model. We provide some computer simulations using an implementation of our scheme to illustrate the validity of the conservation properties.

An Energy-Preserving Scheme for the Coupled Gross-Pitaevskii Equations

An Energy-Preserving Scheme for the Coupled Gross-Pitaevskii Equations

A Lagrangian method for constrained dynamics in tensegrity systems with compressible bars

This paper presents a Lagrangian approach to simulating multibody dynamics in a tensegrity framework with an ability to tackle holonomic constraint violations in an energy-preserving scheme. Governing equations are described using non-minimum coordinates to simplify descriptions of the structure’s kinematics. To minimize constraint drift arising from this redundant system, the direct correction method has been employed in conjunction with a novel energy-correcting scheme that treats the total mechanical energy of the system as a supplementary constraint. The formulation has been extended to allow tensegrity structures with compressible bars, allowing for further discussion on potential choices for softer bar materials. The benchmark example involving a common tensegrity structure demonstrates the superiority of the presented formulation over Simscape Multibody in terms of motion accuracy as well as energy conservation. The effectiveness of the energy correction scheme is found to be increasing with the extent of deformations in the structure.

Coupled analysis to probe the effect of angular assignments on the secondary electron yield (SEY) from copper electrodes

Secondary electron emission from copper is probed utilizing Monte Carlo simulations that take account of elastic scattering based on the Mott theory and inelastic collisions based on energy-dependent energy loss functions. The loss function and stopping power were obtained through first-principles density functional theory. Angular assignment of electrons following elastic scattering or the creation of secondaries is shown to affect the energy-dependent secondary electron yield (SEY). A good match of the simulation results (with a peak SEY of ∼180% at around 300 eV and less than 10% deviation over the 0 to 1000 eV energy range) to available experimental data is shown based on an energy and momentum conservation scheme. Also, the distribution of delay times for the generation of secondaries, the SEY behavior at different incident angles, the energy distribution of emergent secondaries, and their creation profiles as a function of depth are computed to provide a more complete picture of the governing mechanisms and predicted responses.

A Multiobjective Computation Offloading Algorithm for Mobile-Edge Computing

In mobile-edge computing (MEC), smart mobile devices (SMDs) with limited computation resources and battery lifetime can offload their computing-intensive tasks to MEC servers, thus to enhance the computing capability and reduce the energy consumption of SMDs. Nevertheless, offloading tasks to the edge incurs additional transmission time and thus higher execution delay. This article studies the tradeoff between the completion time of applications and the energy consumption of SMDs in MEC networks. The problem is formulated as a multiobjective computation offloading problem (MCOP), where the task precedence, i.e., ordering of tasks in SMD applications, is introduced as a new constraint in the MCOP. An improved multiobjective evolutionary algorithm based on decomposition (MOEA/D) with two performance enhancing schemes is proposed: 1) the problem-specific population initialization scheme uses a latency-based execution location (EL) initialization method to initialize the EL (i.e., either local SMD or MEC server) for each task and 2) the dynamic voltage and frequency scaling-based energy conservation scheme helps to decrease the energy consumption without increasing the completion time of applications. The simulation results clearly demonstrate that the proposed algorithm outperforms a number of state-of-the-art heuristics and metaheuristics in terms of the convergence and diversity of the obtained nondominated solutions.

Mass- and energy-conserving difference schemes for nonlinear fractional Schrödinger equations

In this paper, we present a fully discrete and structure-preserving scheme for the nonlinear fractional Schrödinger equations. The key is to introduce a scalar auxiliary variable and rewrite the equations as a new family of systems. The new systems are approximated by using the implicit midpoint rule, the repeated trapezoidal rule and the fractional centered difference method. It is shown the fully discrete scheme is mass- and energy-conserved. This is sharp contrast to the former result, i.e., the fully discrete scheme is only mass-conserved for the original equations.

Energy Optimal Intelligent Switching Mechanism for Induction Motors with Time Varying Load

In medium and small process industries Induction Motor (IMs) enabled star-delta energy conservation scheme is a common practice. In such industrial applications, the Power Quality Disturbances (PQD) must be considered for hassle free operation. This paper is an attempt in the above direction on star-delta energy conservation applied to IMS (2.2 kW and 37.5 kW) operated with time varying loads. In apart to machine’s performance evaluation, variation of switchover point for uninterrupted operation and optimized energy savings under various electrical perturbations is realized. Moreover, safe restarting feature is equipped in controller for resuming the machine during unplanned shutdown or any interruption.

LINEARLY IMPLICIT ENERGY-PRESERVING FOURIER PSEUDOSPECTRAL SCHEMES FOR THE COMPLEX MODIFIED KORTEWEG–DE VRIES EQUATION

AbstractWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.

A linearly implicit energy-preserving exponential integrator for the nonlinear Klein-Gordon equation

In this paper, we generalize the exponential energy-preserving integrator proposed in the recent paper [SIAM J. Sci. Comput. 38 (2016) A1876–A1895] for conservative systems, which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach. Comparing with the original exponential energy-preserving integrator which usually leads to a nonlinear algebraic system, our new method only involves a linear system with a constant coefficient matrix. Taking the nonlinear Klein-Gordon equation and the nonlinear Schrödinger equation for examples, we derive the concrete energy-preserving schemes and demonstrate their high efficiency through numerical experiments.

Linearly Implicit and High-Order Energy-Conserving Schemes for Nonlinear Wave Equations

A key issue in developing efficient numerical schemes for nonlinear wave equations is the energy-conserving. Most existing schemes of the energy-conserving are fully implicit and the schemes require an extra iteration at each time step and considerable computational cost for a long time simulation, while the widely-used q-stage (implicit) Gauss scheme (method) only preserves polynomial Hamiltonians up to degree 2q. In this paper, we present a family of linearly implicit and high-order energy-conserving schemes for solving nonlinear wave equations. The construction of schemes is based on recently-developed scalar auxiliary variable technique with a combination of classical high-order Gauss methods and extrapolation approximation. We prove that the proposed schemes are unconditionally energy-conserved for a general nonlinear wave equation. Numerical results are given to show the energy-conserving and the effectiveness of schemes.

Compact structure-preserving algorithm with high accuracy extended to the improved Boussinesq equation

The improved Boussinesq equation is numerically studied using a higher-order compact finite difference technique. The aim is to achieve a mass and energy preserving scheme precisely on any time–space regions. The advantage of this scheme is that we can deal with a nonlinear partial differential equation with an implicit linear algorithm. Furthermore, the characteristics of the method are its simple steps and effective clearness. In addition, the convergence and stability analysis are then conducted to search a numerical solution whose the existence and uniqueness are guaranteed. The spatial accuracy is analyzed and found to be fourth order on a uniform grid. The numerical results are compared with established available data in literature for similar test cases, and the results are seen to be in good agreement. Besides, we perform relevant numerical simulations to illustrate the faithfulness of the present method by the evidences of the solitary wave interaction as well as the rapidly depressed solitary waves generation under sufficiently instantly decaying initial data.

A Linearly Implicit Structure-Preserving Scheme for the Camassa–Holm Equation Based on Multiple Scalar Auxiliary Variables Approach

In this paper, we present a linearly implicit energy-preserving scheme for the Camassa–Holm equation by using the multiple scalar auxiliary variables approach, which is first developed to construct efficient and robust energy stable schemes for gradient systems. The Camassa–Holm equation is first reformulated into an equivalent system by utilizing the multiple scalar auxiliary variables approach, which inherits a modified energy. Then, the system is discretized in space aided by the standard Fourier pseudo-spectral method and a semi-discrete system is obtained, which is proven to preserve a semi-discrete modified energy. Subsequently, the linearized Crank–Nicolson method is applied for the resulting semi-discrete system to arrive at a fully discrete scheme. The main feature of the new scheme is to form a linear system with a constant coefficient matrix at each time step and produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution. Several numerical results are addressed to confirm accuracy and efficiency of the proposed scheme.

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrodinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ2 + h4) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

Energy-Efficient Mobile Node in Multicast Batch Verification

Conventional methods used in batch signature verification do not distinguish between static node signature verification and mobile node signature verification. Static nodes rarely face energy expenditure problem, but energy conservation is given a critical importance in mobile nodes. In this paper, we propose a new scheme for energy conservation at mobile receiver. Mobile receiver requests sender to modify the batch size based on the available energy at the receiver; when energy at the mobile receiver has reached particular threshold1, then sender is informed to reduce the batch size to half of the original batch size, which decreases the energy utilized by the mobile receiver for batch signature verification. When the energy at the mobile receiver has reached threshold2, then sender is informed to stop batch transmission till further request from mobile receiver for batch transmission of new batch size; this reduces packet loss which occurs when batches do not reach mobile receiver.

Electromagnetic full- gyrokinetics in the tokamak edge with discontinuous Galerkin methods

We present an energy-conserving discontinuous Galerkin scheme for the full-$f$electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for$\unicode[STIX]{x2202}A_{\Vert }/\unicode[STIX]{x2202}t$, the inductive component of the parallel electric field, using a generalized Ohm’s law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Ampère cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model of the tokamak scrape-off layer using parameters from the National Spherical Torus Experiment (NSTX). This is the first published nonlinear electromagnetic gyrokinetic simulation on open field lines. Comparisons are made to a corresponding electrostatic simulation.

Compact Local Structure-Preserving Algorithms for the Nonlinear Schrödinger Equation with Wave Operator

Combining the compact method with the structure-preserving algorithm, we propose a compact local energy-preserving scheme and a compact local momentum-preserving scheme for the nonlinear Schrödinger equation with wave operator (NSEW). The convergence rates of both schemes are Oh4+τ2. The discrete local conservative properties of the presented schemes are derived theoretically. Numerical experiments are carried out to demonstrate the convergence order and local conservation laws of the developed algorithms.

Energy recovery and conservation utilizing seawater pressure in the working process of Deep-Argo profiling float

Argo profiling float is the key equipment of global ocean observation system. Energy supply issues have limited the working time and the exploring depth of Deep-Argo. In this paper, the scheme of energy recovery and conservation utilizing seawater pressure is put forward innovatively. The hydraulic motor is used to recover and conserve energy when seawater drives hydraulic oil moving from buoyancy bladder to storage bladder in the descending process of Deep-Argo. The working environment of Deep-Argo is defined by the data from sea trials and the simulation model based on AMESim-Simulink is verified by the experiment. Several cases are conducted to analyze the effect of energy recovery and conservation. It shows that the larger fluid drag coefficient of Deep-Argo will lead to the smaller motion speed and the larger energy consumption. The input pressure of hydraulic motor is related with its working time and working frequency. Under the condition that Deep-Argo could reach the target depth, the less volume of hydraulic oil driven by hydraulic pump and the higher input pressure of hydraulic motor will lead to better results of energy recovery and conservation. The recovered energy by the hydraulic motor can be 11.35% of the consumed energy by the hydraulic pump and the conserved energy can be 2.77% of the consumed energy by the hydraulic pump. This paper verifies the feasibility of the scheme for energy recovery and conservation utilizing seawater pressure for Deep-Argo profiling float, and provides the basis for further applied research.

A stable and non-dissipative kinetic energy and entropy preserving (KEEP) scheme for non-conforming block boundaries on Cartesian grids

A stable and non-dissipative numerical scheme for Cartesian methods is proposed. The proposed scheme is based on the second-order kinetic energy and entropy preserving scheme, which the authors proposed recently, and conservation is satisfied at non-conforming boundaries where the grid refinement level is different across the computational block boundaries. In the proposed scheme, ghost cells are used at computational block boundaries for efficient parallel computation, and conservation is satisfied by assigning appropriate values to the ghost cells. Also, although the five conservative variables (i.e., mass, momentum, and total energy) are typically transferred from computational cells to corresponding ghost cells at computational block boundaries, the proposed scheme transfers two more conservative variables to satisfy conservation at non-conforming block boundaries. In a vortex convection test, the convergence rates of the L2- and L∞-error norms are examined, and the proposed scheme preserves the second-order of accuracy without inducing destructive errors at non-conforming boundaries. In an inviscid Taylor-Green vortex simulation, the proposed scheme demonstrates superior numerical stability by preserving kinetic energy and entropy. Also, the proposed scheme performs more stable computations on a non-conforming computational grid than a typical kinetic energy preserving scheme calculated on a uniform computational grid.

Arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation

In this paper, we develop a novel class of arbitrarily high-order energy-preserving schemes for the Camassa-Holm equation. With the aid of the invariant energy quadratization approach, the Camassa-Holm equation is first reformulated into an equivalent system, which inherits a quadratic energy. The new system is then discretized by the standard Fourier pseudo-spectral method, which can exactly preserve the semi-discrete energy conservation law. Subsequently, a symplectic Runge-Kutta method such as the Gauss collocation method is applied for the resulting semi-discrete system to arrive at an arbitrarily high-order fully discrete scheme. We prove that the obtained schemes can conserve the discrete energy conservation law. Numerical results are addressed to confirm accuracy and efficiency of the proposed schemes.

Scalable energy efficient routing mechanism prolonging network lifetime in wireless sensor networks

Due to the tiny size, nodes in a wireless sensor network (WSN) are restricted in battery capacity. As a result, a node and in turn a network of such nodes is not operational for a long time. Further, in some hostile area, battery cannot be replaced easily once deployed. Therefore, an energy conservation scheme or algorithm is needed to prolong the network lifetime. In this paper, an energy efficient routing scheme (EERS) is proposed and evaluated. This scheme extends the lifetime of network by selecting the finest shortest path to transmit data packets to the destined nodes. EERS selects a path/quality link over which it has minimum number of hops with maximum remaining energy over that path. Proposed scheme EERS is compared with other existing schemes for different parameters and varying size of network (1001,000 nodes). The simulation results shows that the EERS scheme prolongs the network lifetime 10%-15% as compared to other considered schemes. EERS also minimises the delay for transmitting data packets and increases the packet delivery ratio.