Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

Structure-preserving algorithms for the Duffng equation

In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.

A structure-preserving algorithm for time-scale non-shiftedHamiltonian systems

The variational calculus of time-scale non-shifted systems includes both the traditional continuous and traditional significant discrete variational calculus.Not only can the combination ofand∇derivatives be beneficial to obtaining higher convergence order in numerical analysis,but also it prompts the timescale numerical computational scheme to have good properties,for instance,structure-preserving.In this letter,a structure-preserving algorithm for time-scale non-shifted Hamiltonian systems is proposed.By using the time-scale discrete variational method and calculus theory,and taking a discrete time scale in the variational principle of non-shifted Hamiltonian systems,the corresponding discrete Hamiltonian principle can be obtained.Furthermore,the time-scale discrete Hamilton difference equations,Noether theorem,and the symplectic scheme of discrete Hamiltonian systems are obtained.Finally,taking the Kepler problem and damped oscillator for time-scale non-shifted Hamiltonian systems as examples,they show that the time-scale discrete variational method is a structure-preserving algorithm.The new algorithm not only provides a numerical method for solving time-scale non-shifted dynamic equations but can be calculated with variable step sizes to improve the computational speed.

Structure-preserving algorithms for autonomous Birkhoffian systems

The Pfaff-Birkhoff variational principle is discretized, and based on the discrete variational principle the discrete Birkhoffian equations are obtained. Taking the discrete equations as an algorithm, the corresponding discrete flow is proved to be symplectic. That means the algorithm preserves the symplectic structure of Birkhofflan systems. Finally, simulation results of the given example indicate that structure-preserving algorithms have great advantage in stability and energy conserving.

LOCAL STRUCTURE-PRESERVING ALGORITHMS FOR THE KLEIN-GORDON-ZAKHAROV EQUATION

In this paper, using the concatenating method, a series of local structure-preserving algorithms are obtained for the Klein-Gordon-Zakharov equation, including four multisymplectic algorithms, four local energy-preserving algorithms, four local momentumpreserving algorithms;of these, local energy-preserving and momentum-preserving algorithms have not been studied before. The local structure-preserving algorithms mentioned above are more widely used than the global structure-preserving algorithms, since local preservation algorithms can be preserved in any time and space domains, which overcomes the defect that global preservation algorithms are limited to boundary conditions. In particular, under appropriate boundary conditions, local preservation laws are global preservation laws.Numerical experiments conducted can support the theoretical analysis well.

LOCAL STRUCTURE-PRESERVING ALGORITHMS FOR THE KDV EQUATION

In this paper, based on the concatenating method, we present a unified framework to construct a series of local structure-preserving algorithms for the Korteweg-de Vries （KdV） equation, including eight multi-symplectic algorithms, eight local energy-conserving algo- rithms and eight local momentum-conserving algorithms. Among these algorithms, some have been discussed and widely used while the most are new. The outstanding advantage of these proposed algorithms is that they conserve the local structures in any time-space re- gion exactly. Therefore, the local structure-preserving algorithms overcome the restriction of global structure-preserving algorithms on the boundary conditions. Numerical experiments are conducted to show the performance of the proposed methods. Moreover, the unified framework can be easily applied to many other equations.

Novel Conformal Structure-Preserving Algorithms for Coupled Damped Nonlinear Schr odinger System

This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schr¨odinger(CDNLS)system,which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws.The proposed algorithms can preserve corresponding conformal multi-symplectic conservation lawand conformalmomentum conservation lawin any local time-space region,respectively.Moreover,it is further shown that the algorithms admit the conformal charge conservation law,and exactly preserve the dissipation rate of charge under appropriate boundary conditions.Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

STRUCTURE-PRESERVING ALGORITHMS FOR DYNAMICAL SYSTEMS

Presents a study which examined the structure-preserving algorithms to phase space volume for linear dynamical systems. Preservation of phase space volume for source-free dynamical systems; Description of a volume-preserving scheme for linear system with canonical form; Information on structure-preserving schemes for linear dynamical systems.

Two Structure-Preserving-Doubling Like Algorithms to Solve the Positive Definite Solution of the Equation X-A^(H)X^(-1)A=Q

In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell(PIC)algorithm in curvilinear orthogonal coordinate systems is developed.The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research.The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms,discrete exterior calculus,and explicit non-canonical symplectic integration.In addition to the truncated infinitely dimensional symplectic structure,the algorithm preserves exactly many important physical symmetries and conservation laws,such as local energy conservation,gauge symmetry and the corresponding local charge conservation.As a result,the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics.The algorithm has been implemented in the Sym PIC code,which is designed for highefficiency massively-parallel PIC simulations in modern clusters.The code has been applied to carry out whole-device 6 D kinetic simulation studies of tokamak physics.A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor.The state also admits a steady-state subsonic ion flow in the range of 10 km s-1,agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated.It is shown that high-n ballooning modes have larger growth rates than low-n global modes,and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2%level by the E×B flow generated by the instability.These results are consistent with early and recent electromagnetic gyrokinetic simulations.

Local structure-preserving methods for the generalized Rosenau-RLW-KdV equation with power law nonlinearity

Local structure-preserving algorithms including multi-symplectic, local energy- and momentum-preserving schemes are proposed for the generalized Rosenau-RLW-KdV equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time-space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.

Structure-preserving properties of Strmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conser- vation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Stormer-Verlet scheme is first constructed in a Hamiltonian frame- work. The conservation law of the StSrmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Stormer-Verlet scheme associated with the con- servation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the StSrmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the StSrmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Stormer-Verlet scheme.

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential （or integro-differential） equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modem mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure- preserving geometric PIC algorithms in comparison with the conventional PIC methods.

Structure-preserving properties of three differential schemes for oscillator system

A numerical method for the Hamiltonian system is required to preserve some structure-preserving properties. The current structure-preserving method satisfies the requirements that a symplectic method can preserve the symplectic structure of a finite dimension Hamiltonian system, and a multi-symplectic method can preserve the multi-symplectic structure of an infinite dimension Hamiltonian system. In this paper, the structure-preserving properties of three differential schemes for an oscillator system are investigated in detail. Both the theoretical results and the numerical results show that the results obtained by the standard forward Euler scheme lost all the three geometric properties of the oscillator system, i.e., periodicity, boundedness, and total energy, the symplectic scheme can preserve the first two geometric properties of the oscillator system, and the St？rmer-Verlet scheme can preserve the three geometric properties of the oscillator system well. In addition, the relative errors for the Hamiltonian function of the symplectic scheme increase with the increase in the step length, suggesting that the symplectic scheme possesses good structure-preserving properties only if the step length is small enough.

Dynamic symmetry breaking and structure-preserving analysis on thelongitudinal wave in an elastic rod with a variable cross-section

The longitudinal wave propagating in an elastic rod with a variable cross-section owns wide engineering background,in which the longitudinal wave dissipation determines some important performances of the slender structure.To reproduce the longitudinal wave dissipation effects on an elastic rod with a variable cross-section,a structure-preserving approach is developed based on the dynamic symmetry breaking theory.For the dynamic model controlling the longitudinal wave propagating in the elastic rod with the variable cross-section,the approximate multi-symplectic form is deduced based on the multi-symplectic method,and the expression of the local energy dissipation for the longitudinal wave propagating in the rod is presented,referring to the dynamic symmetry breaking theory.A structure-preserving method focusing on the residual of the multi-symplectic structure and the local energy dissipation of the dynamic model is constructed by using the midpoint difference discrete method.The longitudinal wave propagating in an elastic rod fixed at one end is simulated,and the local/total energy dissipations of the longitudinal wave are investigated by the constructed structure-preserving scheme in two typical cases in detail.

Underwater four-quadrant dual-beam circumferential scanning laser fuze using nonlinear adaptive backscatter filter based on pauseable SAF-LMS algorithm

The phenomenon of a target echo peak overlapping with the backscattered echo peak significantly undermines the detection range and precision of underwater laser fuzes.To overcome this issue,we propose a four-quadrant dual-beam circumferential scanning laser fuze to distinguish various interference signals and provide more real-time data for the backscatter filtering algorithm.This enhances the algorithm loading capability of the fuze.In order to address the problem of insufficient filtering capacity in existing linear backscatter filtering algorithms,we develop a nonlinear backscattering adaptive filter based on the spline adaptive filter least mean square(SAF-LMS)algorithm.We also designed an algorithm pause module to retain the original trend of the target echo peak,improving the time discrimination accuracy and anti-interference capability of the fuze.Finally,experiments are conducted with varying signal-to-noise ratios of the original underwater target echo signals.The experimental results show that the average signal-to-noise ratio before and after filtering can be improved by more than31 d B,with an increase of up to 76%in extreme detection distance.

Structure-preserving approach for infinite dimensional nonconservative system

The current structure-preserving theory, including the symplectic method and the multisymplectic method, pays most attention on the conservative properties of the continuous systems because that the conservative properties of the conservative systems can be formulated in the mathematical form. But, the nonconservative characteristics are the nature of the systems existing in engineering. In this letter, the structure-preserving approach for the infinite dimensional nonconservative systems is proposed based on the generalized multi-symplectic method to broaden the application fields of the current structure-preserving idea. In the numerical examples,two nonconservative factors, including the strong excitation on the string and the impact on the cantilever, are considered respectively. The vibrations of the string and the cantilever are investigated by the structure-preserving approach and the good long-time numerical behaviors as well as the high numerical precision of which are illustrated by the numerical results presented.

MCWOA Scheduler:Modified Chimp-Whale Optimization Algorithm for Task Scheduling in Cloud Computing

Cloud computing provides a diverse and adaptable resource pool over the internet,allowing users to tap into various resources as needed.It has been seen as a robust solution to relevant challenges.A significant delay can hamper the performance of IoT-enabled cloud platforms.However,efficient task scheduling can lower the cloud infrastructure’s energy consumption,thus maximizing the service provider’s revenue by decreasing user job processing times.The proposed Modified Chimp-Whale Optimization Algorithm called Modified Chimp-Whale Optimization Algorithm(MCWOA),combines elements of the Chimp Optimization Algorithm(COA)and the Whale Optimization Algorithm(WOA).To enhance MCWOA’s identification precision,the Sobol sequence is used in the population initialization phase,ensuring an even distribution of the population across the solution space.Moreover,the traditional MCWOA’s local search capabilities are augmented by incorporating the whale optimization algorithm’s bubble-net hunting and random search mechanisms into MCWOA’s position-updating process.This study demonstrates the effectiveness of the proposed approach using a two-story rigid frame and a simply supported beam model.Simulated outcomes reveal that the new method outperforms the original MCWOA,especially in multi-damage detection scenarios.MCWOA excels in avoiding false positives and enhancing computational speed,making it an optimal choice for structural damage detection.The efficiency of the proposed MCWOA is assessed against metrics such as energy usage,computational expense,task duration,and delay.The simulated data indicates that the new MCWOA outpaces other methods across all metrics.The study also references the Whale Optimization Algorithm(WOA),Chimp Algorithm(CA),Ant Lion Optimizer(ALO),Genetic Algorithm(GA)and Grey Wolf Optimizer(GWO).

Enhancing Cancer Classification through a Hybrid Bio-Inspired Evolutionary Algorithm for Biomarker Gene Selection

In this study,our aim is to address the problem of gene selection by proposing a hybrid bio-inspired evolutionary algorithm that combines Grey Wolf Optimization(GWO)with Harris Hawks Optimization(HHO)for feature selection.Themotivation for utilizingGWOandHHOstems fromtheir bio-inspired nature and their demonstrated success in optimization problems.We aimto leverage the strengths of these algorithms to enhance the effectiveness of feature selection in microarray-based cancer classification.We selected leave-one-out cross-validation(LOOCV)to evaluate the performance of both two widely used classifiers,k-nearest neighbors(KNN)and support vector machine(SVM),on high-dimensional cancer microarray data.The proposed method is extensively tested on six publicly available cancer microarray datasets,and a comprehensive comparison with recently published methods is conducted.Our hybrid algorithm demonstrates its effectiveness in improving classification performance,Surpassing alternative approaches in terms of precision.The outcomes confirm the capability of our method to substantially improve both the precision and efficiency of cancer classification,thereby advancing the development ofmore efficient treatment strategies.The proposed hybridmethod offers a promising solution to the gene selection problem in microarray-based cancer classification.It improves the accuracy and efficiency of cancer diagnosis and treatment,and its superior performance compared to other methods highlights its potential applicability in realworld cancer classification tasks.By harnessing the complementary search mechanisms of GWO and HHO,we leverage their bio-inspired behavior to identify informative genes relevant to cancer diagnosis and treatment.

Rao Algorithms-Based Structure Optimization for Heterogeneous Wireless Sensor Networks

The structural optimization of wireless sensor networks is a critical issue because it impacts energy consumption and hence the network’s lifetime.Many studies have been conducted for homogeneous networks,but few have been performed for heterogeneouswireless sensor networks.This paper utilizes Rao algorithms to optimize the structure of heterogeneous wireless sensor networks according to node locations and their initial energies.The proposed algorithms lack algorithm-specific parameters and metaphorical connotations.The proposed algorithms examine the search space based on the relations of the population with the best,worst,and randomly assigned solutions.The proposed algorithms can be evaluated using any routing protocol,however,we have chosen the well-known routing protocols in the literature:Low Energy Adaptive Clustering Hierarchy(LEACH),Power-Efficient Gathering in Sensor Information Systems(PEAGSIS),Partitioned-based Energy-efficient LEACH(PE-LEACH),and the Power-Efficient Gathering in Sensor Information Systems Neural Network(PEAGSIS-NN)recent routing protocol.We compare our optimized method with the Jaya,the Particle Swarm Optimization-based Energy Efficient Clustering(PSO-EEC)protocol,and the hybrid Harmony Search Algorithm and PSO(HSA-PSO)algorithms.The efficiencies of our proposed algorithms are evaluated by conducting experiments in terms of the network lifetime(first dead node,half dead nodes,and last dead node),energy consumption,packets to cluster head,and packets to the base station.The experimental results were compared with those obtained using the Jaya optimization algorithm.The proposed algorithms exhibited the best performance.The proposed approach successfully prolongs the network lifetime by 71% for the PEAGSIS protocol,51% for the LEACH protocol,10% for the PE-LEACH protocol,and 73% for the PEGSIS-NN protocol;Moreover,it enhances other criteria such as energy conservation,fitness convergence,packets to cluster head,and packets to the base station.