Second-order Runge-Kutta Method Research Articles

Coherent Chaotic Communication Using Generalized Runge–Kutta Method

Computer simulation of continuous chaotic systems is usually performed using numerical methods. The discretization may introduce new properties into finite-difference models compared to their continuous prototypes and can therefore lead to new types of dynamical behavior exhibited by discrete chaotic systems. It is known that one can control the dynamics of a discrete system using a special class of integration methods. One of the applications of such a phenomenon is chaos-based communication systems, which have recently attracted attention due to their high covertness and broadband transmission capability. Proper modulation of chaotic carrier signals is one of the key problems in chaos-based communication system design. It is challenging to modulate and demodulate a chaotic signal in the same way as a conventional signal due to its noise-like shape and broadband characteristics. Therefore, the development of new modulation–demodulation techniques is of great interest in the field. One possible approach here is to use adaptive numerical integration, which allows control of the properties of the finite-difference chaotic model. In this study, we describe a novel modulation technique for chaos-based communication systems based on generalized explicit second-order Runge–Kutta methods. We use a specially designed test bench to evaluate the efficiency of the proposed modulation method and compare it with state-of-the-art solutions. Experimental results show that the proposed modulation technique outperforms the conventional parametric modulation method in both coverage and noise immunity. The obtained results can be efficiently applied to the design of advanced chaos-based communication systems as well as being used to improve existing architectures.

Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay.

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge-Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.

THE DERIVED EXPLICIT FOURTH-STAGE SECOND-ORDER RUNGE-KUTTA METHOD IS CONSISTENT AND CONVERGENT

The purpose of this paper is to analyze the consistency and convergence of an explicit fourth-stage second-order Runge Kutta method derived using Taylor series expansion by varying parameters. The analysis revealed that the method is consistent and convergent. These properties are key and vital for any numerical method to possess for it to be capable of handling initial value problems. The implementation of this method on initial-value problems has been done in previous paper, and it revealed that the method compared favorably well with the other existing explicit Runge Kutta method of the same order. Hence, as a result of the convergence and consistency of the method, it will definitely be reliable and dependable.

Mathematical model to investigate transmission dynamics of COVID-19 with vaccinated class

<abstract><p>The susceptible, exposed, infected, quarantined and vaccinated (SEIQV) population is accounted for in a mathematical model of COVID-19. This model covers the therapy for diseased people as well as therapeutic measures like immunization for susceptible people to enable understanding of the dynamics of the disease's propagation. Each of the equilibrium points, i.e., disease-free and endemic, has been proven to be globally asymptotically stable under the assumption that $ \mathscr{R}_0 $ is smaller or larger than unity, respectively. Although vaccination coverage is high, the basic reproduction number depends on the vaccine's effectiveness in preventing disease when $ \mathscr{R}_0 > 0 $. The Jacobian matrix and the Routh-Hurwitz theorem are used to derive the aforementioned analysis techniques. The results are further examined numerically by using the standard second-order Runge-Kutta (RK2) method. In order to visualize the global dynamics of the aforementioned model, the proposed model is expanded to examine some piecewise fractional order derivatives. We may comprehend the crossover behavior in the suggested model's illness dynamics by using the relevant derivative. To numerical present the results, we use RK2 method.</p></abstract>

A high-resolution upwind numerical scheme for solving transient multiphase flow in wellbore

The identification of gas-liquid interface is the basis for accurate calculation of wellbore flow parameters after gas invasion, which plays an important role in the safety and reliability of drilling engineering. At present, most petroleum engineering uses a simple difference algorithm to calculate the wellbore multiphase flow, this scheme can not capture the gas-liquid interface accurately, and the numerical accuracy is low. Therefore, this paper proposes a mathematical and numerical calculation model for transient multiphase flow in wellbore during gas invasion. Using second-order Runge-Kutta method and OMUSCL2 method, a high-precision and high-resolution upwind AUSMV numerical scheme is constructed to capture the gas-liquid interface. The research shows that the high-precision and high-resolution upwind AUSMV scheme can accurately capture the gas-liquid interface, and thus accurately solve the multiphase flow parameters in wellbore during the gas invasion.

An efficient and accurate interpolation method for parametric curve machining.

A subsection interpolation method based on the curve curvature threshold is proposed to resolve the incompatible problem of machining accuracy and machining efficiency in parametric curve machining. In the pre-interpolation stage, the curve curvature threshold is calculated based on geometric and kinematic constraints. The subsection interpolation key points and their nominal velocities are then determined from the curvature threshold points and the start and end points of the curve, and the arc length of each subsegment can be calculated based on the adaptive Simpson method. As a result, the S-type speed planning algorithm and the bidirectional speed scanning algorithm are used to update and realize the global speed curve to reduce the speed fluctuation. In the real-time interpolation stage, the curve interpolation parameters are calculated using the parametric modified second-order Runge–Kutta method, which could improve the interpolation accuracy significantly and also shorten the interpolation time. Finally, it is found using numerical cases that the proposed method can smooth the overall interpolation speed, reduce the speed fluctuation effectively and improve the real-time performance of the interpolation.

Balanced Meshless Method for Numerical Simulation of Pollutant Transport by Shallow Water Flow over Irregular Bed: Application in the Strait of Gibraltar

This paper focuses on the implementation of an accurate meshless scheme for the simulation of the advection–diffusion of non-active pollutant in a two-dimensional depth-averaged flow. The depth-averaged flow model includes the shallow water system and the pollutant propagation described by the advection–diffusion equation with diffusion tensor. The mathematical model was implemented by using a meshless method based on local radial basis functions. This method was used to numerically evaluate the spatial derivatives and complemented with the second-order Runge–Kutta method for the time evolution. To remove the non-physical oscillations, which appear at the discontinuity, a filter based on a hyperviscosity operator was applied. To investigate the effectiveness and accuracy of the proposed scheme, a number of tests are presented, including the dam break problem, and the pure transport of a pollutant in a long channel. Finally, a hypothetical example of a pollutant transport in the Strait of Gibraltar is modeled. The results obtained are compared both with analytical solutions and with simulation results obtained by a finite volume method based on the Roe-MUSCL scheme. The main advantages of the proposed method are: (i) the simplicity of implementation, (ii) the ability to handle calculations of slowly varying flows or concentrations, as well as rapidly varying flows containing shocks or discontinuities, and (iii) the ability to satisfy the C-property and guarantee positive values of both water level and pollutant concentration.

Meshless Generalized Finite Difference Method for the Propagation of Nonlinear Water Waves under Complex Wave Conditions

The propagation of nonlinear water waves under complex wave conditions is the key issue of hydrodynamics both in coastal and ocean engineering, which is significant in the prediction of strongly nonlinear phenomena regarding wave–structure interactions. In the present study, the meshless generalized finite difference method (GFDM) together with the second-order Runge–Kutta method (RKM2) is employed to construct a fully three-dimensional (3D) meshless numerical wave flume (NWF). Three numerical examples, i.e., the propagation of freak waves, irregular waves and focused waves, are implemented to verify the accuracy and stability of the developed 3D GFDM model. The results show that the present numerical model possesses good performance in the simulation of nonlinear water waves and suggest that the 3D “RKM2-GFDM” meshless scheme can be adopted to further simulate more complex nonlinear problems regarding wave–structure interactions in ocean engineering.

Numerical Investigation of Contact Burning in an Air-Breathing Continuous Rotating Detonation Engine

Three-dimensional (3D) numerical simulations of a continuous rotating detonation engine are carried out with an unsteady Reynolds-averaged Navier-Stokes solver. The second-order upwind advection upstream splitting method and second-order Runge-Kutta method are used to discretize space and time terms, and detailed 9-species 19-step hydrogen-oxygen reactions are applied in this study. Nonpremixed rotating detonation is successfully realized numerically, and the characteristics of the detonation wave are revealed. The expanding angle of the combustor has a great impact on the shape of the detonation wave but has little influence on the propagation velocity. The evolution of combustion on the contact region is analyzed in detail; a more accurate schematic of non-premixed air-breathing rotating detonation engines is given in this paper. A rough analysis of the heat performance of the contact region shows that the heat release of the contact region is approximately one-third of the total heat release and the configurations of the combustors do not affect the proportion.

Повышение производительности контурной механической обработки путем сглаживания пространственной кусочно-линейной траектории и квазиоптимального планирования подач

Currently, not all control systems of multi-axis machine tools and industrial robots are capable of efficiently machining the products with a complex surface shape. The toolpath for such products is usually described by blocks using G01 operator frames, which are short segments. To machine such a contour, it is necessary to periodically decrease the feed rate at the conjugation points of the segments, which leads to a decrease in performance and a lower quality of machining. The study introduces a method to solve the problem: smoothing the toolpath by including spline segments in it. When smoothing, we used a cubic B-spline with five control points, which made it possible for the entire path to have geometric continuity G2. The smoothing scheme enabled us to analytically express the maximum curvature, take into account the specified error in the approximation of the spline construction and the mutual intersection of adjacent curves. Contour machining performance was improved by using a bidirectional frame preview algorithm, which accounted for the specified geometric constraints and chordal spline construction error, as well as limits of the contour rate, acceleration and jerk in each segment of the path. The second-order Runge-Kutta method with a compensation approximation scheme was used for parametric interpolation, which made it possible to reduce fluctuations in the feed rate and positively affected the quality of the machined product surface. The experimental results confirm the correctness of the chosen approach and its validity in the control system for high-speed machining of products with complex surfaces.

Impact of difference between explicit and implicit second-order time integration schemes on isotropic/anisotropic steady incompressible turbulence field

This study presents the impact of the difference between the implicit and explicit time integration methods on a steady turbulent flow field. In contrast to the explicit time integration method, the implicit time integration method may produce significant kinetic energy conservation error because the widely used spatial difference method for discretizing the governing equations is explicit with respect to time. In this study, the second-order Crank-Nicolson method is used as the implicit time integration method, and the fourth-order Runge-Kutta, second-order Runge-Kutta and second-order Adams-Bashforth methods are used as explicit time integration methods. In the present study, both isotropic and anisotropic steady turbulent fields are analyzed with two values of the Reynolds number. The turbulent kinetic energy in the steady turbulent field is hardly affected by the kinetic energy conservation error. The rms values of static pressure fluctuation are significantly sensitive to the kinetic energy conservation error. These results are examined by varying the time increment value. These results are also discussed by visualizing the large scale turbulent vortex structure.

Formulative Visualization of Numerical Methods for Solving Non-Linear Ordinary Differential Equations

Many physical problems in the real world are frequently modeled by ordinary differential equations (ODEs). Real-life problems are usually non-linear, numerical methods are therefore needed to approximate their solution. We consider different numerical methods viz., Explicit (Forward) and Implicit (Backward) Euler method, Classical second-order Runge-Kutta (RK2) method (Heun’s method or Improved Euler method), Third-order Runge-Kutta (RK3) method, Fourth-order Runge-Kutta (RK4) method, and Butcher fifth-order Runge-Kutta (BRK5) method which are popular classical iteration methods of approximating solutions of ODEs. Moreover, an intuitive explanation of those methods is also be presented, comparing among them and also with exact solutions with necessary visualizations. Finally, we analyze the error and accuracy of these methods with the help of suitable mathematical programming software.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

A meshless generalized finite difference method for solving shallow water equations with the flux limiter technique

In this study, a novel meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations. Since they form a hyperbolic system of equations, discontinuous solutions are allowed to transmit during the simulation. The generalized finite difference-split coefficient matrix method, recently proposed, is applied and improved using the flux limiter to eliminate the possible-appearing numerical oscillations. In the proposed scheme, the split-coefficient matrix method is adopted to convert the shallow water equations to the characteristic form. Then, the generalized finite difference method and the second-order Runge-Kutta method are employed for spatial and temporal discretization, respectively. The upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method. Applying the flux limiter technique, the expressions can automatically switch the proper discrete order when facing discontinuous solutions. Although the limiter function required the derivatives of different orders, the generalized finite difference method can solve these necessary expressions using the first- and second-order Tayler series. Several numerical examples are provided to demonstrate the capability of the proposed scheme, and the results are compared with other numerical schemes to show the effectiveness of the proposed generalized finite difference-flux limiter method.

Dynamically Consistent NSFD Methods for Predator-prey System

In this paper, we introduce two nonstandard finite difference (NSFD) methods for solving the mathematical model of the Rosenzweig-MacArthur predator-prey system. These new proposed numerical methods have important features such as positivity and elementary stability. Numerical comparisons between the proposed methods and the other methods such as second-order and forth order Runge-Kutta methods (we refer them RK2 and RK4, respectively), Euler method, and NSFD method presented in [6] indicate that the new methods have better accuracy and convergence.

Numerical Simulating Open-Channel Flows with Regular and Irregular Cross-Section Shapes Based on Finite Volume Godunov-Type Scheme

A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

Numerical Study of Detonation Processes in Rotating Detonation Engine and its Propulsive Performance

Abstract Numerical studies on detonation wave propagation in rotating detonation engine and its propulsive performance with one- and multi-step chemistries of a hydrogen-based mixture are presented. The computational codes were developed based on the three-dimensional Euler equations coupled with source terms that incorporate high-temperature chemical reactions. The governing equations were discretized using Roe scheme-based finite volume method for spatial terms and second-order Runge-Kutta method for temporal terms. One-dimensional detonation simulations with one- and multi-step chemistries of a hydrogen-air mixture were performed to verify the computational codes and chemical mechanisms. In two-dimensional simulations, detonation waves rotating in a rectangular chamber were investigated to understand its flowfield characteristics, where the detailed flowfield structure observed in the experiments was successfully captured. Three-dimensional simulations of two-waved rotating detonation engine with an annular chamber were performed to evaluate its propulsive performance in the form of thrust and specific impulse. It was shown that rotating detonation engine produced constant thrust after the flowfield in the chamber was stabilized, which is a major difference from pulse detonation engine that generates repetitive and intermittent thrust.

On the propagation of nonlinear water waves in a three-dimensional numerical wave flume using the generalized finite difference method

Nonlinear water waves are common physical phenomena in the field of coastal and ocean engineering, which plays a critical role in the investigation of hydrodynamics regarding offshore and deep-water structures. In the present study, a three-dimensional (3D) numerical wave flume (NWF) is constructed to simulate the propagation of nonlinear water waves. On the basis of potential flow theory, the second-order Runge-Kutta method (RKM2) combining with a semi-Lagrangian approach is carried out to discretize the temporal variable of the 3D Laplace’s equation. For the spatial variables, the generalized finite difference method (GFDM) is adopted to solve the governing equations for the deformable computational domain at each time step. The upstream condition is considered as a wave-making boundary with imposing horizontal velocity while the downstream condition as a wave-absorbing boundary with a pre-defined sponge layer to deal with the phenomenon of wave reflection. Three numerical examples are investigated and discussed in detail to validate the accuracy and stability of the developed 3D GFDM-based NWF. The results show that the newly-proposed numerical method has good performance in the prediction of the dynamic evolution of nonlinear water waves, and suggests that the novel 3D “RKM2-GFDM” meshless scheme can be employed to further investigate more complicated hydrodynamic problems in practical applications.

Wave-Structure Interaction for a Stationary Surface-Piercing Body Based on a Novel Meshless Scheme with the Generalized Finite Difference Method

The wave-structure interaction for surface-piercing bodies is a challenging problem in both coastal and ocean engineering. In the present study, a two-dimensional numerical wave flume that is based on a newly-developed meshless scheme with the generalized finite difference method (GFDM) is constructed in order to investigate the characteristics of the hydrodynamic loads acting on a surface-piercing body caused by the second-order Stokes waves. Within the framework of the potential flow theory, the second-order Runge-Kutta method (RKM2) in conjunction with the semi-Lagrangian approach is carried out to discretize the temporal variable of governing equations. At each time step, the GFDM is employed to solve the spatial variable of the Laplace’s equation for the deformable computational domain. The results show that the developed numerical method has good performance in the simulation of wave-structure interaction, which suggests that the proposed “RKM2-GFDM” meshless scheme can be a feasible tool for such and more complicated hydrodynamic problems in practical engineering.

Exponential Time Differencing Algorithm for Pulse-Coupled Hodgkin-Huxley Neural Networks.

The exponential time differencing (ETD) method allows using a large time step to efficiently evolve stiff systems such as Hodgkin-Huxley (HH) neural networks. For pulse-coupled HH networks, the synaptic spike times cannot be predetermined and are convoluted with neuron's trajectory itself. This presents a challenging issue for the design of an efficient numerical simulation algorithm. The stiffness in the HH equations are quite different, for example, between the spike and non-spike regions. Here, we design a second-order adaptive exponential time differencing algorithm (AETD2) for the numerical evolution of HH neural networks. Compared with the regular second-order Runge-Kutta method (RK2), our AETD2 method can use time steps one order of magnitude larger and improve computational efficiency more than ten times while excellently capturing accurate traces of membrane potentials of HH neurons. This high accuracy and efficiency can be robustly obtained and do not depend on the dynamical regimes, connectivity structure or the network size.