Runge-Kutta Research Articles

Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System

To study the nonlinear dynamic behavior and system stability of a rubbing overhung rotor with viscoelastic and memory-effect damping and random uncertain parameters, this paper introduces a fractional-order modeling and stochastic dynamic analysis method for the nonlinear overhung rotor system with frictional impact faults. Firstly, the dynamic equations of the overhung rotor considering friction effect and fractional damping effect are established based on the transfer matrix method and fractional order derivative. Then, the time-domain response of the fractional-order dynamic equations is solved by combining the Runge–Kutta method and the continuous fractional expansion, and the steady-state response characteristics of different fractional damping are analyzed in the deterministic case. Finally, to analyze the response of the system under the effect of stochastic parameters, the sparse grid-based PCE metamodel of the system response is developed. Statistical moments, probability distributions, and sensitivity indices of the response of stochastic systems are revealed. The results of this paper provide a theoretical basis for efficient and accurate prediction of the stochastic response of nonlinear rubbing overhung rotor systems.

Dynamic analysis of a high-speed micro compressor supported by spiral-grooved hydrodynamic air bearing

Spiral-grooved hydrodynamic air bearings are used in high-speed machines due to their advantages of oil-free, low friction, and compact structure. This paper aims to investigate the dynamic characteristics of a high-speed rotor supported by spiral-grooved hydrodynamic air bearings and the effects of spiral groove parameters. A mathematical model of the rotor-bearing system is developed by coupling the rotor nonlinear motion equations and the bearing dynamic Reynolds equation. The finite element method combined with the Runge-Kutta method is employed simultaneously to obtain the nonlinear trajectory of the rotor-bearing system. The theoretical model is verified by experiments on the built micro compressor prototype. The numerical predicted onset speed of the sub-synchronous motion of the rotor-bearing system agrees well with the experimental observations. Furthermore, the effects of radius clearance, groove depth, groove angle, groove axial and circumferential width on the dynamic performance of the rotor-bearing system are analyzed. The results show that radius clearance has the most significant influence, followed by groove depth, groove axial, and then circumferential width. The groove angle has the least impact.

Physics-informed neural networks approach for pollutant dispersion flow caused by magnetic dipole

Abstract Research on the liquid flow and distribution of waste discharge concentration over cylindrical surfaces might lead to the growth of efficient waste management and pollution controlling strategies. In view of this, the pollutant concentration and magnetic dipole impact on the liquid flow through the cylinder with a porous medium are explored in the present analysis. Moreover, Fourier’s and Fick’s laws are considered to explore heat transport features. The governing partial differential equations (PDEs) of the present study are converted into non-dimensional ordinary differential equations (ODEs) utilizing similarity variables. Also, the current study proposes a novel implementation of neural networks based on physics knowledge for the liquid flow past a cylinder. The self-updating weights and bias facilitate the network to satisfy the minimization objective function and provide accurate results. Additionally, the solution obtained by Runge Kutta Fehlberg's fourth-fifth order (RKF-45) method is utilized to validate the physics-informed neural network (PINN) results, presenting mean squared errors from 10-9 to 10-11. Furthermore, it is found that the increase in temperature relaxation time parameter decreases the Nusselt number gradually at the rate of 1 to 5%. An upsurge in the values of the pollutant external source parameter and the pollutant external source variation parameter decreases the concentration profile.

Thermal transportation of radiative cathode nanotube‐based nanofluid on a stretching/shrinking wedge with varying magnetism and heat source

AbstractThe complex heat transfer processes occurring in a magnetized water‐based hybrid nanofluid flowing past a stretching or shrinking wedge are investigated in this work. The research highlights the significant influence of the wedge angle on fluid flow patterns and temperature gradients, emphasizing how larger angles cause changes in pressure, flow separation, and temperature profiles. Additionally, the study emphasizes the importance of radiation effects near surfaces, where increased radiation often leads to cooling due to thermal energy emission. The findings have implications in industries such as aerospace, electronics, thermal management, and groundwater resources management where precise temperature control is critical for safe and efficient operations. The numerical method applied to solve the governing equations in the study is the fourth‐order Runge–Kutta technique with shooting technique. This method efficiently solves the boundary value problem by converting it into a system of first‐order initial value problems. Then shooting technique is used to solve the system, starting with an adequate initial guess. The study's results are presented in tabular and graphical forms, providing valuable insights into the complex fluid dynamics and thermal behavior in the given setting, enabling more informed decision‐making and optimization in relevant engineering applications. Increased radiation frequently results in a cooling effect owing to thermal energy output. Heat sinks/heat sources are crucial in temperature regulation and are effective in lowering temperatures via processes. We also emphasize the importance of radiation impacts near surfaces, where increased radiation frequently leads to cooling owing to thermal energy emission. The numerical values of skin friction and Nusselt number for various parameters in the cases of shrinking and stretching provide crucial insights into the heat transmission and frictional properties of the system under examination. These quantitative results contribute to a better understanding of the complex fluid dynamics and thermal behavior in the given setting, emphasizing the significance of the research findings.

Output performance analysis of a piezoelectric micro nuclear battery powered by radioisotopes

Abstract This paper investigates the output performance (output voltage, electrical energy output, power output) of a piezoelectric micro nuclear battery powered by radioisotopes. The theoretical formulations are base on Euler-Bernoulli beam theory and include the effects of load nonlinearity due to radioactive source. By employing extended Hamilton’s principle, the nonlinear electromechanical Lagrange equations are derived then solved by using Runge-Kutta method to obtain the dynamic output response. The results based on present electromechanical dynamic model are validated through direct comparisons with the results in the open literatures. The effects of initial gap, length of the piezoelectric layer, thickness of the collector and load impedance on the output voltage, energy output and power output of the piezoelectric micro nuclear battery are discussed in detail.

Comparison of Runge-Kutta Methods for Solving Nonlinear Equations

This research explores the numerical solutions of nonlinear ordinary differential equations using four distinct methods: Heun method, the Midpoint method, Ralston's method, and the fourth-order Runge-Kutta method (RK4). Given the complexity and prevalence of nonlinear equations in various scientific fields, effective numerical techniques are essential for obtaining accurate solutions. This research contributes to the understanding of numerical methods in solving complex differential equations, aiding practitioners in selecting the most appropriate approach for their needs.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Modelling Of The Weed Invasion And Control Strategies For Mesquite Plants

The new study presents an innovative model for examining the spread of the mesquite species, which is well-known for its extensive presence around the world thanks to water dispersion in various areas. The goal is to assess the stability of the system locally and globally using epidemiological modeling techniques. The approach involved analyzing the basic reproduction number (R₀) using the next-generation matrix to find the coefficients for the Lyapunov function. The results indicated that, the absence of mesquite remains stable on both local and global scale when R₀ is less than 1, but when R₀ exceeds 1, an endemic equilibrium is observed. Numerical simulations using Runge Kutta ode45 algorithm implemented in MATLAB. The study demonstrated that, an increase in the number of water bodies flowing on land, leads to a faster spread and invasion of mesquite. The study suggests the use of this model in areas where the mesquite species is native to raise awareness, inform decision-making, and educate communities, ultimately aiding efforts to manage and eliminate the species.

Global Analysis of Meningitis Disease With Optimal Control

The meningitis epidemic has impacted lives negatively, especially in Sub-Sahara Africa, dubbed the ‘Meningitis Belt’. The epidemic has been a public health concern due to an improper understanding of the disease’s dynamics. To implement a control measure that will help minimize the epidemic, we introduce a non-linear Meningitis model that describes the dynamic behaviour of the disease and explains the transmission trend. The model explores the condition that leads to local or global asymptomatic stability of the equilibria. The model is subjected to a sensitivity analysis to find the parameters that influence the R0. The model is modified into an optimal control by adding timedependent controls. The control model is solved qualitatively using Pontryagin’s maximum principle and numerically using MATLAB and the fourth-order Runge-Kutta method. We provide a control strategy that can be relied on for management decision-making based on the results.

Induced magnetic field and thermal regulation of synovial fluid flow through irregular surfaces with first-order slip and gold nanoparticles

ABSTRACT The current computational exploration develops a poroelastic model of the knee cartilage to elevate its temperature under cyclic sinusoidal loading. A contributing factor to the rise in temperature is this tissue’s viscous dissipation, which converts some of the mechanical energy supplied into heat. The manipulation of cartilage temperature is mostly reliant on the movement of synovial fluid inside the space between joints and within the permeable cartilage. We developed the region flow model in the presence of induced magnetic fields and gold nanoparticles, which enhanced the efficiency of joint lubrication by boosting the load-carrying capacity. We dealt with the problem for two different models. Model 1 assumes that viscosity is exponentially dependent on concentration. The shear thinning index in Model 2 is regarded as a concentration-dependent variable. We introduce these models for a complex wavy surface, where the first-order slip effect occurs. We modified the governing problem by assuming a long wavelength and a low Reynolds number. We followed up with a computational analysis utilizing the Rung–Kutta–Merson approach with Newton iteration in a shooting and matching strategy. We have conducted detailed comparisons between Model 1 and Model 2. The graphical findings have been shown for the distributions of velocity, temperature, and nanoparticle concentration. Additionally, the pressure gradients, streamlines, and axially generated magnetic fields have also been included. These results are presented for numerous physical parameters. The mechanics of trapping are thoroughly examined through the use of streamlines.

Reversible esterification process in Casson nanofluids: A numerical analysis of non-newtonian hydromagnetic flow

A computational mechanism has been formulated to scientifically analyze the phenomenon of combined convection in the occurrence of magnetohydrodynamic flow behavior of Casson nanofluid. This study emphasizes the trajectory of the nanofluid along a stretching sheet with nonlinear permeability, while considering additional physical effects such as reversible chemical reaction, thermal radiation, energy source/sink, suction and viscous dissipation. In this study, we have additionally integrated the nanofluid paradigm proposed by Buongiorno, which encompasses the repercussion of thermophoresis along with Brownian motion. The utilization of appropriate similarity renovations serves the purpose of recasting the dominant multivariate differential equations to a collection of nonlinear differential equations in single variable. The Runge–Kutta shooting mechanism is implemented for the intention of numerically determining the unknown boundary conditions. The solution to the dominant equations was obtained by employing the fourth-order Runge–Kutta technique and to ensure the dependability of the outcomes, the bvp4c method was employed. The graphical and tabulated observations are presented herein to facilitate a comprehensive analysis of the underlying physical characteristics inherent in the problem at hand. The utilization of the Casson parameter has been found to be advantageous in the reduction of shear stress rate, along with the enhancement of mass and energy transfer rates. Additionally, the application of suction has been observed to be beneficial in the enhancement of Sherwood number and energy transfer rate. Within the scope of the context of the esterification process, the purpose of this proposal is to evaluate the impact that numerous arithmetical values exert on the temperature field, the velocity profile and the volumetric concentration. An in-depth graphical analysis that takes into account the relevant physical consequences is used to evaluate the most important factors that relate to the physical quantities that describe the contours of temperature, velocity and concentration. In addition to being accompanied by their respective explanations, the tabular portrayal of the local Sherwood number, shear stress rate along with local Nusselt number all feature in this paper. There is a clear distinction that can be made between irreversible and reversible flows in the evaluation of the local Sherwood number, rate of shear stress and local Nusselt number. This differentiation is brought about by taking into analysis thermophoresis parameter, Brownian motion parameter and suction parameter. The results that were produced from the theoretical simulations have significant repercussions for a variety of fields that are related to energy engineering. The findings derived from the analysis indicate a negative correlation between the Brownian motion parameter and both irreversible and reversible flow in terms of rate of energy transfer and shear stress rate. It is imperative to highlight that reversible flow holds greater significance in comparison to irreversible flow.

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge–Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 4$$\\end{document}. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

Linearly Implicit Conservative Schemes for the Nonlocal Schrödinger Equation

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes aim to significantly enhance computational accuracy and efficiency, while maintaining the essential conservative properties necessary for accurate physical modeling. This offers a structured approach to handle auxiliary variables, ensuring stability and conservation, while the symplectic Runge–Kutta method provides a robust framework with high accuracy. Together, these techniques offer a powerful and reliable approach for researchers dealing with complex quantum mechanical systems described by the nonlocal Schrödinger equation, ensuring both accuracy and stability in their numerical simulations.

Topological search and gradient descent boosted Runge–Kutta optimiser with application to engineering design and feature selection

Topological search and gradient descent boosted Runge–Kutta optimiser with application to engineering design and feature selection

Shock wave propagation in a real gas with or without gravitational field in the presence of magnetic field and monochromatic radiation via group invariance method

PurposeThis article aims to find the similarity solutions for the one-dimensional motion of spherical symmetric shock wave in non-ideal gas influenced by the azimuthal magnetic field and monochromatic radiation in the presence or absence of gravitational field. This paper also aims to study the effects of physical parameters on the strength of shock wave, and on the flow variables in the flow-field region behind the shock front.Design/methodology/approachThe Roche model is used to describe the gravitational field effects due to a massive nucleus at the point of symmetry. To derive the similarity solutions, the Lie group symmetry method has been used. Also, the numerical solutions to the present problem are obtained by using Rung–Kutta method of the fourth order with the use of Mathematica software. The effects of variation in the parameter of non-idealness of the gas, the gravitation parameter, the strength of the ambient magnetic field and the adiabatic index of the gas on the shock wave, and on the flow variables is discussed. A comparative study between with and without gravitational field is also, made.FindingsFor different choices of the arbitrary constants that appeared in the solution of infinitesimal generators, we have obtained seven distinct cases of similarity solutions. In the absence of the gravitational field, the similarity solution exists to the power and exponential law shock paths, but in the presence of gravitational field, the similarity solution exists to the power law shock path case only. In the absence of gravitational field, the shock strength is enhanced in the exponential law shock path case in comparison to the power law shock path case. It is found that the shock wave decays with an increase in the value of the adiabatic exponent, the strength of magnetic field, non-idealness of the gas or gravitational parameter.Research limitations/implicationsThe consideration of medium under the influence of gravitational field due to a heavy nucleus at the center and presence of magnetic field decrease the shock strength. This result may be helpful in designing space vehicle and jet engine.Practical implicationsThe result of the present study may be used in the analysis of data from the measurements by space craft in the solar wind and in neighborhood of the Earth’s magnetosphere.Social implicationsThe obtained results may be used for mankind.Originality/valueThe study of spherical shock wave propagation influenced by monochromatic radiation and azimuthal magnetic field in a non-ideal gas with or without gravitational field has yet to be discussed by any authors by using the Lie group symmetry method. In this article, we have discussed all possible cases of similarity solutions using the Lie group symmetry method, which is not studied by anyone as known to us.

Spectro-spatial analysis of van der Pol-type phononic crystals

Abstract The application of phononic chains as metamaterials demonstrates their remarkable capability to manipulate the propagation of waves. These periodic structures yield frequency-dependent behavior of material comprising characteristics with many possible engineering applications. In this paper, we investigate the weak and general nonlinear behaviors of the van der Pol-type damped phononic chains. The analysis of wave propagation is initially conducted for a one-dimensional structure, and subsequently, is extended to consider the wave motion through two-dimensional and three-dimensional lattices. Results are obtained using the method of multiple scales and a Spectro-spatial analysis by employing the numerical method of the 4th-order Runge–Kutta. A new phase-diagram relation within the chain’s unit cell is also introduced aiming to enhance the numerical findings. Our results indicate that in the weakly nonlinear regime, the van der Pol-type damping closely follows the linear dispersion curve, regardless of the initial amplitude. This suggests a symmetry between energy pumping and dissipation modes, where hardening and softening behaviors align with linear characteristics of common damping mechanisms, such as viscous damping. Additionally, the formulation demonstrates the existence of limit-cycle stability in the motion of each mass. For the general damped system, it is observed that a special frequency exists where the system converges, for all wave numbers similar to the synchronization effect. Hence, the motion and the frequency of all masses are synced. Additionally, non-reciprocal wave propagation is observed, resulting in a bandgap structure with a symmetry breaking occurring near the limit cycle. These results are promising in the fields of wave emitters, wave filters, and signal encryption.

Study of MHD Casson Nanofluid Flow Over an Inclined Stretching Sheet in Porous Media with Thermal Slip, Chemical Reaction and Radiation Effects

In the current study, the behaviour of Casson nanofluid subjected to magnetohydrodynamic (MHD) flow across an inclined stretched sheet within a porous medium has been investigated numerically. The governing equations are transformed into ordinary differential equations with the corresponding boundary conditions by employing similarity transformations. The solutions of essential equations are achieved by using the 4th-order Runge-Kutta method combined with the shooting technique. The novelty and innovative contribution are showcased through illustrative graphs that scrutinize the effect of factors that impact the velocity, temperature, and concentration profiles within assorted flow scenarios. The primary focal points of the study encompass examining variations in magnetic field strength, angle of inclination, and suction intensity that affect the fluid's velocity moderation, while improved porosity and radiation parameters lead to a rise in fluid temperature. Higher Biot numbers correlate with an increase in fluid temperature. The implications of positive coefficients of heat transfer are crucial across various fields to ensure efficient thermal management ensuring optimal performance and longevity. The numerical data presented is aligned with earlier published results for comparison. Furthermore, the variations in skin friction, Nusselt, and Sherwood numbers driven by different parameters are displayed in tables to highlight significant modifications.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.