Fourth-order Scheme Research Articles

Nonlinear vibration analysis of geometrically imperfect FGM Timoshenko beams with porosity and elastically restrained ends in thermal environments

This paper aims to analyze the combined influences of pore imperfection, initial geometric imperfection, elasticity of tangential constraints of ends, elastic foundations and elevated temperature on the nonlinear vibration of functionally graded material (FGM) beams. The pores are assumed to distribute into FGM according to even and uneven patterns and effective properties of porous FGM are estimated using a modified rule of mixture. Motion equations of geometrically imperfect beams are established within the framework of Timoshenko beam theory incorporating von Kármán nonlinearity and foundation interaction. Analytical solutions are assumed to satisfy simply supported and clamped conditions of beam ends and Galerkin method is applied to derive a nonlinear time ordinary differential equation. The frequencies of nonlinear free vibration are determined employing fourth-order Runge-Kutta numerical integration scheme. Parametric studies are carried out to assess numerous effects on both linear and nonlinear frequencies. The results reveal that frequency nonlinearity is more significant for beams with tangentially restrained ends, especially at elevated temperatures. It is also found that geometric imperfection increases the linear frequency and lowers ratio of nonlinear-to-linear frequencies.

Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate

Abstract Casson fluids containing carbon nanotubes of various lengths and radii on a moving permeable plate reduce friction and improve equipment efficiency. They improve plate flow dynamics to improve heat transfer, particularly in electronic cooling and heat exchangers. The core objective of this study is to investigate the heat transmission mechanism and identify the prerequisites for achieving high cooling speeds within a two-dimensional, stable, axisymmetric boundary layer. This study considers a sodium alginate-based nanofluid containing single/multi-wall carbon nanotubes (SWCNTs/MWCNTs) and Casson nanofluid flow on a permeable moving plate with varying length, radius, and nonlinear thermal radiation effects. The plate has the capacity to move either parallel to or perpendicular to the free stream. The governing partial differential equations for the boundary layer, which are interconnected, are transformed into standard differential equations. These equations are then numerically solved using the Runge–Kutta fourth-order scheme incorporated in the shooting method. This research analyses and graphically displays the effects of factors including mass suction, nanoparticle volume fraction, Casson parameter, thermal radiation, and temperature ratio. Additionally, a comparison is made between the present result and the previous finding, which presented in a tabular format. The coefficient of skin friction decreases in correlation with an increase in Casson fluid parameters and Prandtl number. Heat transfer rate decreases with a variation in viscosity parameter, while it is increasing with an increase in Prandtl number. In addition, this study demonstrates that heat transfer rate for MWCNT is significantly higher than that of SWCNT nanoparticles. Thermal radiation and temperature ratio reduce the heat transfer rate, whereas nanoparticle volume fraction and Casson parameter enhance it over a shrinking surface.

A higher order numerical scheme to a nonlinear McKendrick–Von Foerster equation with singular mortality

In this paper, higher-order numerical schemes to the McKendrick–Von Foerster equation are presented when the death rate has singularity at the maximum age. The third, fourth-order schemes that are proposed here are based on the characteristics which are non-intersecting lines in this case, and are multi-step methods with appropriate corrections at each step. In fact, the convergence analysis of these schemes is discussed in detail. Moreover, numerical experiments are provided to validate the orders of convergence of the proposed third-order and fourth-order schemes.

A parallel multi-resolution Smoothed Particle Hydrodynamics model with local time stepping

Smoothed Particle Hydrodynamics (SPH) model has exhibited remarkable efficacy in addressing strongly nonlinear large deformation problems. However, the expensive computational cost could limit its application. To address this, we propose a novel parallel multi-resolution SPH model with a local time step strategy. This model integrates the multi-resolution model with a Message Passing Interface-based parallelization framework, wherein subdomains discretize the computational domain at varying resolutions. Meanwhile, a local fourth-order Runge-Kutta time integration scheme is introduced, enabling different subdomains to use different time steps. Subdomains of varying resolutions are devoid of overlapping regions, and changes in particle resolution at interfaces are achieved through a dynamic strategy involving particle merging and splitting. Besides that, we conducted a comparative analysis of different SPH discretization formats concerning their impact on the accuracy of multi-resolution particle discretization. Finally, several numerical tests were conducted to validate the accuracy and efficiency of the present multi-resolution SPH model.

2DH numerical model of nonlinear wave propagation based on a hybrid finite-volume and finite-difference scheme

A two-dimensional hybrid (2DH) numerical model suitable for nonlinear wave propagation from deep to shallow waters is developed based on a hybrid finite-volume and finite-difference scheme. To utilize this method conveniently, the selected governing equations are derived again in a conservative form. The fourth-order monotonic upstream-centered scheme for conservation laws–total variation diminishing scheme with a Riemann solver is employed to calculate the numerical flux. The finite-difference scheme is employed to discretize the other spatial derivatives, and the third-order strong stability-preserving (SSP) Runge–Kutta scheme is used for time-stepping. Wave breaking is simulated by locally switching the governing equations to nonlinear shallow-water equations when the Froude number exceeds a certain threshold. A comparison of the numerical results and analytical or experimental data showed that the numerical model performs well in simulating nearshore wave propagation.

Dynamics of Hepatitis B Virus Disease with Infectious Latent and Vertical Transmission

Hepatitis B has become a major health threat because it is a life-threatening liver disease with an estimated 0.25 billion people suffering from this infectious disease worldwide. This paper presents a SLITR (Susceptible-Latent-Infectious-Treatment-Recovery) mathematical model that combines both vaccination and treatment as a means of controlling the hepatitis B virus (HBV). The nonlinear ordinary differential equations for the HBV transmission capacities were resolved and the basic reproduction number R0 computed using the next generation matrix method and simulated numerically using the Runge-Kutta fourth order scheme implemented using MatLab. The stability points for disease-free equilibrium state (DFE), endemic equilibrium state (EE), and basic reproduction number R0 were obtained and the results show that the disease-free equilibrium was both locally and globally asymptotically stable (R0<1) . Similarly, treatment or vaccine administered was effective in alleviating the spread of HBV disease, and when both control strategies are combined, the diseases are quickly controlled and eventually eradicated.

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

Slab waveguide communication study using Finite Difference Method (FDM) with fourth-order compact scheme

An accurate and efficient semi-vectorial mode solver, examining Electric (E) and Magnetic (H) fields, is utilized to explore the communication characteristics of various slab waveguides. This approach relies on the Finite Difference Method (FDM) using uniform grid points, employing a Higher (Fourth) Order Compact type technique coupled with Conjugate Gradient (CG) iteration. The present study is an extension of the previous work where various aspects of E- and H-field propagations through rib-strucured waveguide was analyzed successfully. By incorporating the refractive index profile of the slab structure, excellent functionality of the waveguide is demonstrated here in Electric field by utilizing the concepts of normalized index and effective or modal index, confinement factor, and the modal birefringence properties which proves to be critical for potential applications in photonic integrated circuits. The materials chosen for the purpose are AlGaAs-GaAs and GeSi which play a vital role in characterizing the waveguides’performances through simulation using MATLAB. The use of surface and contour plots aids in visualizing the distribution of corresponding field within the guided modes. This analysis significantly contributes in understanding the waveguide's behavior and confinement capability during the field propagation. Most importantly, the variations of the waveguides’performance indices with their structure parameters help to identify their optimized values for fabrication to enhance the transmission efficiency of the waveguide.

Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach

It is suggested that an explicit-implicit third-order in-time numerical scheme be used to solve time-dependent partial differential equations (PDEs). A fourth-order compact scheme in space is adopted for discretizing space variables. The work creatively blends a third-order explicit-implicit numerical scheme in time with a fourth-order compact scheme in space to solve time-dependent PDEs. Notably, the proposed numerical scheme demonstrates stability for scalar PDEs and exhibits convergence for both linear and nonlinear PDE systems. In addition, a mathematical model is given for the heat transfer of magnetohydrodynamic (MHD) flow over flat and oscillatory sheets, and the effect of viscous dissipation is also considered. For various governing parameters, features of the flow and heat transfer characteristics are discussed and examined. Some results are verified using an exact solution. We use an explicit-implicit finite difference method guaranteed to be stable and convergent under all conditions to solve the coupled nonlinear partial differential equations that describe this problem. Physical aspects of the problem are discussed in light of the system’s embedded parameters, and numerical and graphical results are presented. To sum up, progress in our knowledge of such systems relies heavily on creating an accurate and efficient numerical scheme for modeling MHD boundary layer flows while considering the influence of viscous dissipation. This study aims to improve the current state of MHD flow simulations and provide a useful resource for engineers and scientists engaged in developing and improving MHD-based technologies.

Fast nonlinear Fourier transform algorithms for optical data processing.

The nonlinear Fourier transform (NFT) is an approach that is similar to a conventional Fourier transform. In particular, NFT allows to analyze the structure of a signal governed by the nonlinear Schrödinger equation (NLSE). Recently, NFT applied to NLSE has attracted special attention in applications of fiber-optic communication. Improving the speed and accuracy of the NFT algorithms remains an urgent problem in optics. We present an approach that allows to find all variants of symmetric exponential splitting schemes suitable for the fast NFT (FNFT) algorithms with low complexity. One of the obtained schemes showed good numerical results in computing the continuous spectrum compared with other fast fourth-order NFT schemes.

Viscoelastic fluid flow on channel in porous medium with Soret and Dufour along with the effect of activation energy and suction and blowing

This study conducts a thorough numerical analysis of a complex problem involving magnetohydrodynamic flow, radiation, activation energy, Soret, the Dufour effect, and heat transfer in a permeable channel for a Maxwell fluid. Governing equations for momentum, concentration, and heat are transformed into ordinary differential equations through a similarity transformation, enabling focused analysis. Utilizing a fourth-order Runge-Kutta scheme, we compute dimensionless velocity, temperature, and concentration fields for the steady motion of a Maxwell fluid in a channel. To ensure accuracy, our results are validated against existing research, showing excellent agreement. The study investigates the impact of various physical parameters on fluid motion, revealing insights such as decreased velocity and concentration with increased chemical reaction, and rising fluid temperature. Additionally, concentration distribution decreases with a chemical reaction and a higher Schmidt number. This research enhances understanding of complex fluid dynamics, providing valuable insights for engineering applications, particularly in industries requiring precise fluid behavior control.

A Five-Step Block Method Coupled with Symmetric Compact Finite Difference Scheme for Solving Time-Dependent Partial Differential Equations

In this article, we present a five-step block method coupled with an existing fourth-order symmetric compact finite difference scheme for solving time-dependent initial-boundary value partial differential equations (PDEs) numerically. Firstly, a five-step block method has been designed to solve a first-order system of ordinary differential equations that arise in the semi-discretisation of a given initial boundary value PDE. The five-step block method is derived by utilising the theory of interpolation and collocation approaches, resulting in a method with eighth-order accuracy. Further, characteristics of the method have been analysed, and it is found that the block method possesses A-stability properties. The block method is coupled with an existing fourth-order symmetric compact finite difference scheme to solve a given PDE, resulting in an efficient combined numerical scheme. The discretisation of spatial derivatives appearing in the given equation using symmetric compact finite difference scheme results in a tridiagonal system of equations that can be solved by using any computer algebra system to get the approximate values of the spatial derivatives at different grid points. Two well-known test problems, namely the nonlinear Burgers equation and the FitzHugh-Nagumo equation, have been considered to analyse the proposed scheme. Numerical experiments reveal the good performance of the scheme considered in the article.

Effects of MHD and thermal radiation on non-Newtonian fluid with Cattaneo–Christov heat over a stretching flow surface

In this study, we are investigating the flow of an incompressible Williamson fluid using a Cattaneo–Christov heat flux model, with consideration for a magnetic Reynolds number that is not high. This fluid flows over a linearly stretched 2-D surface and is affected by a magnetic field, thermal radiation-diffusion, and heat generation. To model the Williamson flow, we apply a boundary layer approximation and develop the ordinary differential equations. The system of differential equations is transformed into a system of ordinary differential equations using a suitable transformation. We utilize the Iterative Runge–Kutta fourth-order scheme to numerically solve the boundary conditions associated with the nonlinear, dimensionless ordinary differential equations. Graphical representations of the results are employed to analyze the impact of physical properties on velocity, temperature, and concentration, with the numerical results presented in tabular form. Upon comparing our findings with previously published results, we have found them to be in good agreement. The velocity decreases as the Williamson fluid factor gradually increases. Williamson fluids are recognized for their shear thinning behavior, wherein viscosity decreases as the shear rate increases. However, higher parameter values can lead to increased fluid viscosity, potentially reducing the degree of shear thinning. This phenomenon results in a more uniform viscosity profile across varying shear rates, contributing to a reduced velocity distribution within the fluid.

Numerical Solution of Fourth-order Initial Value Problems Using Novel Fourth-order Block Algorithm

In this paper, a one-step fourth-order block scheme for solving fourth order Initial Value Problems (IVPs) of Ordinary Differential Equations (ODE) is developed using interpolation and collocation techniques. The derived schemes contain two hybrid points which are chosen such that 0 < w1 < w2 < 1 where w1 and w2 are defined as hybrid points. The characteristics of the developed schemes are analyzed. The obtained schemes are applied in block form to solve some fourth-order IVPs and the numerical results show the accuracy and effectiveness of the block scheme compared with some existing methods.

A modified Chebyshev collocation method for the generalized probability density evolution equation

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

Nonlinear dynamical characteristics of carbon nanotube-reinforced composite beams with piezoelectric actuators and elastically restrained ends under thermo-electro-mechanical loads

Nonlinear free vibration and dynamical responses of carbon nanotube (CNT) reinforced composite beams with surface-bonded piezoelectric layers and tangentially restrained ends under thermo-electro-mechanical loads are investigated in this paper. The properties of constitutive materials are assumed to be temperature-dependent and effective properties of nanocomposite are estimated using an extended rule of mixture. Unlike previous studies, the present work considers the effects of tangentially elastic constraints of two ends on the nonlinear dynamic characteristics of hybrid beams. Motion equation is established within the framework of Euler-Bernoulli beam theory taking into account von Kármán nonlinearity. Analytical solution is assumed to satisfy simply supported boundary conditions and Galerkin procedure is employed to obtain a time ordinary differential equation including both quadratic and cubic nonlinear terms. This differential equation is numerically solved employing fourth-order Runge-Kutta scheme to determine the frequencies of nonlinear free vibration and nonlinear transient response. Parametric studies are executed to examine numerous influences on the nonlinear dynamical characteristics of hybrid nanocomposite beams. The study reveals that tangential constraints of ends substantially effect the frequencies and dynamic response of the beam, especially at elevated temperatures. The results also indicate that nonlinear dynamic responses can be controlled effectively by means of piezoelectric actuators and elasticity of tangential constraints of ends should be considered in design of piezo-CNTRC beams.

Efficient parametric family of fourth-order Jacobian-free iterative vectorial schemes

AbstractIn this work, a multiparametric family of iterative vectorial fourth-order methods free of Jacobian matrices is proposed. A convergence analysis of this family is carried out as well as a study of its efficiency. Several numerical experiments are made in order to compare the behaviour of the proposed family with other competitive methods of the literature.

Pulsatile flow of Casson hybrid nanofluid between ternary-hybrid nanofluid and nanofluid in an inclined channel with temperature-dependent viscosity

This study presents a mathematical model for layered non-miscible liquid flow in an inclined channel subject to a pulsatile pressure gradient. The Casson hybrid nanofluid (hnf) is sandwiched between the layers of Newtonian ternary hybrid nanofluid (t-hnf) and nanofluid (nf). Additionally, the impact of temperature-dependent viscosity is taken into consideration. Multiwall carbon nanotubes, molybdenum disulfide, and silver are considered nanoparticles, with water and kerosene oil as base fluids. The governing partial differential equations are transformed into coupled non-linear ordinary differential equations (ODEs) using the perturbation technique. The resulting ODEs are solved numerically using MATHEMATICA software with the help of the shooting technique associated with the Runge-Kutta fourth-order scheme. The behaviors of numerous sundry parameters against velocity, temperature, concentration, and shear stress are depicted graphically and in tabular form. The outcomes of the study reveal that if the viscosity is held constant, then the fluid’s velocity is initially low and gradually increases as the viscosity variation parameter rises. The velocity of fluid flow is higher when a Newtonian fluid occupies the middle region of the channel than when Casson fluid does. The increase in the nanoparticle shape factor, at both boundaries, results in an increase in the rate of heat transfer.

Near-critical behavior of the Zhong-Zhang model.

The Zhong-Zhang (ZZ) model is a one-degree-of-freedom dynamical system describing the motion of an insulating plate of length d floating on the upper surface of a convecting fluid, with locking at the boundaries. In the absence of noise, the system away from the boundaries is described by linear differential equationswith a delay time τ. The d,τ plane consists of two domains separated by a critical curve. For asymptotically long times, subcritical orbits approach a nontrivial periodic attractor, while the supercritical ones tend to a stationary state at the origin. We investigate near-critical behavior using a modified fourth-order Runge-Kutta integration scheme. We then construct a piecewise analytic decomposition of the periodic attractor, which makes possible a far higher level of accuracy. Our results provide solid evidence for an asymptotic power-law approach to criticality of several observables. The power laws are fed back to determine the piecewise-analytic structure deep into the near-critical regime. In an Appendix, we explore the effect of introducing noise using modified order-3/2 Kloeden-Platen-Schurz stochastic integration, following several observable quantities through the near-critical parameter domain.

A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers’ equation

This study explores a third-order backward differentiation formula (BDF3) and nonlinear fourth-order difference method (FODM) for solving the generalized viscous Burgers’ equation (GVBE). The BDF3 method is employed for discretizing the time derivative, while the nonlinear term uλux is handled using a newly constructed nonlinear fourth-order difference operator. The spatial second derivative is discretized using a linear fourth-order difference formula. We establish the convergence of the proposed BDF3-FODM by introducing a cut-off function (COF). The main contributions include the construction of a novel nonlinear fourth-order difference operator, addressing the limitation observed in Zhang et al. (2021), Guo et al. (2023), where the order of spatial convergence was restricted to second order. Finally, a numerical test is presented to validate our theoretical analysis.