Fifth-order Method Research Articles

Two-Step Fifth-Order Efficient Jacobian-Free Iterative Method for Solving Nonlinear Systems

This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments that confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Computational study of Carreau nanofluid across a stretching cylinder by homogeneous-heterogeneous responses: A frame work of modified Buongiorno's nanofluid model

This paper studies the Carreau nanofluid employed by modified Buongiorno's model to theoretically analyze a homogeneous-heterogeneous process-influenced numerical solution of a Carreau nanofluid flowing through a stretched cylinder. In steady of Carreau nanofluid model proposed by Buongiorno's flow across a linearly extending cylinder influenced by homogeneous/heterogeneous responses are examined and express the changes of moment, energy, and concentration in both graphical and numerical. The following impact factors are consequences of the result, the stretching limiting factor, the curvature limiting factor and other limiting factors. The effects of both convection and heat generation are seen in flow field analysis. In comparison to ambient fluid, the stipulated exterior temperature and concentration are hypothetically increased. For the regulating equations of velocity, energy, and concentration profiles are permuting from a set of partial differential equations stand to ordinary differential equations manipulating an equivalent conversion. The resulting sequence of equations was solved at the boundary value problem in the fifth-order method. The velocity, temperature, concentration, skin friction coefficient, local Nusselt number, and local Sherwood number fields are studied, and then the outputs are illustrated in graphs and tables.

Impact of chemical reactions that generate and absorb heat in the flow induced by a squeezing porous slider

The three-dimensional flow of a viscous fluid induced by an expanding or contracting porous slider under the influence of activation energy with exothermic and endothermic chemical reactions is explored in this study. Moreover, the amount of fluid injected to levitate the slider changes over time according to where it is at any instant. With the aid of similarity variables, the modelling equations relating to the fluid flow are converted into a system of ordinary differential equations. Then, this system of equations is solved numerically with the help of the Runge–Kutta Fehlberg’s fourth fifth-order method (RKF-45). Graphs are used to analyze the impact of the various parameters on the flow, thermal and concentration fields. Results reveal that the velocity profiles get smaller as the wall dilation parameter rises. An anticipated boundary layer development next to the wall results from increased Reynolds number. The temperature profile for an exothermic process has a diminishing influence as the activation energy parameter increases, whereas the opposite consequences are obtained for an endothermic reaction. For an exothermic reaction, the temperature profile rises as the chemical reaction parameter values increase. However, the opposite consequences can be seen for an endothermic reaction.

An approach to the effects of stellar rotation on the theoretical apsidal motion constants. Calculations from 0.40 M_sol to 25.0 M_sol

The most reliable sources for determining absolute stellar parameters are the double-lined eclipsing binary systems. Some of these systems also show apsidal motion, characterized by the variable log k$_2$. This point grants us the possibility of investigating the stellar interior, specifically the degree of stellar mass concentration. The first studies carried out about four decades ago on this topic showed appreciable discrepancies not only with respect to the comparison between the observed absolute dimensions and their theoretical counterparts, but mainly concerning the degree of mass concentration through the analysis of their apsidal motions. Fortunately, this scenario has been gradually improving with the advances in the quality of observational techniques and advances in the input physics of the evolutionary stellar models (e.g. opacities, thermonuclear reactions, equations of state, numerical techniques). These new developments in the input physics has improved the comparison between observations and the values predicted by theory, including the apsidal motion rates. This progress has lead us to investigate second-order effects such as rotation and dynamic tides. In this paper we deal with the effects of rotation on the degree of mass concentration. The stellar models were calculated using the MESA package. The mass range studied here was 0.40 to 25.0 M$_ odot $ for a generic chemical composition characterized by X = 0.70 and Z = 0.02. The present models were computed without taking into account core overshooting in order to highlight the effects of rotation. Each model was followed from the pre-main sequence until the central hydrogen content is of the order of or less than 0.03, covering the range of masses and evolutionary status of the double-lined eclipsing binaries showing apsidal motion. Regarding the calculation of the internal structure coefficients bf we integrated the Radau equation using the fifth-order Runge-Kutta method and a tolerance level of 10$^ $. As an auxiliary tool, homology transformations have been used to explain, qualitatively, how the equation of state, thermonuclear reactions, and convection impact the degree of mass concentration of the models. In our past studies on the effects of rotation on log k$_2$ an average correction was used for all models together. For this a small range of masses (2.0, 7.0, 15.0 M$_ odot $) has been used. In the present paper the corrections due to the effects of rotation in log k$_2$ are presented for each mass individually and for three evolutionary phases. This point is particularly important bf given that these corrections show a clear dependence on the mass and on the evolutionary status. Such corrections can be easily introduced into the theoretical calculation of apsidal motion rates through a linear equation. A typical correction due to the rotation effects is of the order of -0.03 in log k$_2$.

An efficient fifth-order block method for solving third-order BVPs

This research paper introduces a new block method to solve third-order boundary-value problems (BVPs) directly. The method uses a suitable polynomial and three intermediate points to approximate the theoretical solution of third-order BVPs. An approximate solution is provided within the integration interval, and the basic properties of the proposed method, including convergence and stability, are theoretically analyzed. To assess the effectiveness of the proposed method, several real-life problems, such as boundary layer problems and physical model problems, are solved numerically. After comparison with some existing methods, the results of numerical experiments confirm the practical utility and effectiveness of the proposed method in real-world applications by demonstrating that it is superior and highly efficient in performance.

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

<abstract><p>Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.</p></abstract>

SEIRS model for malaria transmission dynamics incorporating seasonality and awareness campaign

Malaria, a devastating disease caused by the Plasmodium parasite and transmitted through the bites of female Anopheles mosquitoes, remains a significant public health concern, claiming over 600,000 lives annually, predominantly among children. Novel tools, including the application of Wolbachia, are being developed to combat malaria-transmitting mosquitoes. This study presents a modified susceptible-exposed-infectious-recovered-susceptible (SEIRS) compartmental mathematical model to evaluate the impact of awareness-based control measures on malaria transmission dynamics, incorporating mosquito interactions and seasonality. Employing the next-generation matrix approach, we calculated a basic reproduction number (R0) of 2.4537, indicating that without robust control measures, the disease will persist in the human population. The model equations were solved numerically using fourth and fifth-order Runge-Kutta methods. The model was fitted to malaria incidence data from Kenya spanning 2000 to 2021 using least squares curve fitting. The fitting algorithm yielded a mean absolute error (MAE) of 2.6463 when comparing the actual data points to the simulated values of infectious human population (Ih). This finding indicates that the proposed mathematical model closely aligns with the recorded malaria incidence data. The optimal values of the model parameters were estimated from the fitting algorithm, and future malaria dynamics were projected for the next decade. The research findings suggest that social media-based awareness campaigns, coupled with specific optimization control measures and effective management methods, offer the most cost-effective approach to managing malaria.

A General Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

A family of fifth-order iterative methods is proposed for solving nonlinear equations using the weight function technique. This family offers flexibility through its structure and the choice of weight functions, resulting in a wide range of new specific schemes. It is demonstrated that this proposed family includes several well-known and recent methods as special cases. In addition, several new particular methods are designed to achieve better results than existing methods of the same type. Convergence analysis is conducted, and numerical examples in both real and complex domains are provided for several specific schemes within the proposed family. Comparisons between the existing methods within this family and the newly introduced methods generally indicate improved performance among the new members. Notably, the study of complex dynamics and basins of attraction reveals that our new specific schemes have broader sets of initial points that lead to convergence.

On a Fifth-Order Method for Multiple Roots of Nonlinear Equations

There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods.

Hydrothermal analysis of Ag and CuO hybrid NPs suspended in mixture of water 20%+EG 80% between two concentric cylinders

This paper investigates the mixture of hybrid nanofluid (NF) flow between two concentric cylinders numerically. Ethylene glycol (80%) and water (20%) comprise the base fluid. Additionally, copper oxide (CuO) and silver (Ag) NPs are added. The hybrid NF flows between the two coaxial cylinders is studied. The inner cylinder is fixed and the outer cylinder is rotating. A radial magnetic field also affects fluid flow, and joule heating joule heating is also considered. The transformed system of nonlinear ordinary differential equations is solved numerically using the fifth-order Runge-Kutta method. Numerical solutions are obtained for fluid velocity and velocity gradient, fluid temperature and temperature gradient, pressure, surface friction coefficient and Nusselt number under the influence of different variables. The computer code is validated by comparing the variation of temperature of hybrid nanofluid with the numerical published data, a good agreement was observed. As the magnetic field parameter is increased, the fluid's velocity reduces. The fluid temperature increases as the magnetic field parameter and Brinkman number were increased. The skin friction coefficient increases with the enhancement of volume percentage and the magnetic field parameter. The Nusselt number rises with increasing of the magnetic field and Brinkman number.

Effect of Nanoparticle Diameter in Maxwell Nanofluid Flow with Thermophoretic Particle Deposition

The time-dependent Maxwell nanofluid flow with thermophoretic particle deposition is examined in this study by considering the solid–liquid interfacial layer and nanoparticle diameter. The governing partial differential equations are reduced to ordinary differential equations using suitable similarity transformations. Later, these reduced equations are solved using Runge–Kutta–Fehlberg’s fourth and fifth-order method via a shooting approach. An artificial neural network serves as a surrogate model, making quick and precise predictions about the behaviour of nanofluid flow for various input parameters. The impact of dimensionless parameters on flow, heat, and mass transport is determined via graphs. The results reveal that the velocity profile drops with an upsurge in unsteadiness parameter values and Deborah number values. The rise in space and temperature-dependent heat source/sink parameters value increases the temperature. The concentration profile decreases as the thermophoretic parameter upsurges. Finally, the method’s correctness and stability are confirmed by the fact that the maximum number of values is near the zero-line error. The zero error is attained near the values 2.68×10−6, 2.14×10−9, and 8.5×10−7 for the velocity, thermal, and concentration profiles, respectively.

A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number ≈ 1 ), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

Local Refinement and Adaptive Strategy for a System of Free Boundary Power Options with High Order Compact Differencing

In this research, we propose fourth-order non-uniform Hermitian differencing with a fifth-order adaptive time integration method for pricing system of free boundary exotic power put options consisting of the option value, delta sensitivity, and gamma. The main objective for implementing the above scheme is to carefully account for the irregularity in the locality of the left corner point after fixing the free boundary. Specifically and mainly, we stretch the performance of our proposed method threefold. First, we exploit the non-uniform fourth-order Hermitian scheme to locally concentrate space grid points arbitrarily close to the left boundary. Secondly, we further leverage the adaptive nature of the embedded time integration method, which allows optimal selection of a time step based on the space grid point distribution and regional variation. Thirdly, we introduce a fourth-order combined Hermitian scheme, which requires fewer grid points for computing the near boundary point of the delta sensitivity and gamma. Another novelty is how we approximate the optimal exercise boundary and its derivative using a fifth-order Robin boundary scheme and fourth-order combined Hermitian scheme. Our proposed method consistently achieves reasonable accuracy with very coarse grids and little runtime across the numerical experiments. We further compare the results with existing methods and the ones we obtained from the uniform space grid.

Bioconvection on Non-Newtonian Magnetohydrodynamics Carreau Nanofluid with Activation Energy and Binary Chemical Reaction in Darcy Forchhiemer Porous Medium

In this present research, suspensions of gyrotactic micro-organisms on non-newtonian carreau nanofluid with activation energy and binary chemical reaction is examined which is implanted under Darcy Forchhiemer porous medium. Similarity transformations are used to convert the governing equation into a highly nonlinear ODE’s. Fifth-order Runge Kutta-Fehlberg method using shooting technique is utilized to solve ODE’s. Velocity, temperature, concentration, and density of motile gyrotactic microorganisms profile were depicted graphically and explained briefly. It is observed that activation energy enhances the concentration profile of both pseudoplastic and dilatant fluids. Present results is compared with previously published work which is found to be in good agreement.

Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package

Rosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family.

High-Order Accurate Solutions of the Point Kinetics Equations with the Spectral Deferred Correction Method

Solving initial value problems with high-order methods receives considerable attention in many fields because these methods can potentially improve the accuracy of the simulation results with lower computational cost than low-order methods. Most methods, however, are either complicated to implement or unstable when the order of accuracy is high. The spectral deferred correction (SDC) method is a stable, robust, and efficient high-order time-integration scheme capable of an arbitrary order of accuracy. In this paper, we apply the SDC method to solve the initial value problem of the point kinetics equations (PKEs). For our implementation, we show that SDC is -stable for orders up to eight and the order of accuracy is verified for PKE problems with a range of different reactivities. A fifth-order SDC method was then implemented to solve the exact PKE in the transient multilevel method of MPACT. The error from solutions of the exact PKE with SDC is shown to be negligible. The investigations made here can provide the foundation for future investigations simulating the neutron transport problem using the high-order methods for both spatial discretization and time integration.

Thermal performance analysis of magnetized convectively heated ferromagnetic nanoparticles for radiative cross nanofluid flow

The main purpose of this work is to investigate the innovation of flow of two-dimensional (2D) ferrofluid flow passing upon stretched surface having the effect of a magnetic dipole. The said problem is selected since its use is very common in biomedical as well as engineering applications together with systems of drug targeting under magnetic effects. The impacts of magnetic dipole along with radiation parameter in flow of ferromagnetic cross fluid for a stretched region are investigated in detail by use of suitable similarity variables the nonlinear, ODEs are obtained. The desired ODEs are resolved numerically by using RK fifth-order method parallel to shooting scheme. Impacts of ferromagnetic interaction, viscous dissipation, Curie temperature, and Buongiorno parameters with convective boundary conditions are perceived for velocity, temperature and concentration fields. Furthermore, the terms like velocity, mass transfer and thermal gradients are under consideration and pictorial scrutinizing is done. Moreover, temperature of ferromagnetic fluid upsurges for enlargement in the values of ferrohydrodynamics interaction and Curie temperature parameters. Concentration of ferrofluid reveals a contrary tendency against the parameters of thermophoresis and Brownian motion. Owing to extensive application prospects of magnetized nanofluids, this area has received interest from engineers and researchers; nanofluids have a noteworthy preference over ordinary fluids.

General solution of phase mismatched second harmonic conversion efficiency in periodically poled nonlinear medium

In this work we provide for the first time a novel and more accurate second harmonic generation (SHG) conversion efficiency equation for periodically poled nonlinear medium. We believe the equation derived in this work is more useful than equations reported in previous works which are limited because it is considering the case of either nondepleted pump regime or perfect phase matching. The equation derived in this work tackles the general case: depleted pump regime and phase mismatching which makes it important for theoretical – experimental study. We used MgO:PPLN crystal as an example to illustrate SHG conversion efficiency as function of crystal length, laser power density, crystal temperature and beam divergence. We included fundamental and second harmonic waves reflection at entrance and exit surfaces respectively and performed numerical solution using fifth-order Runge-Kutta method. The numerical results show excellent match with analytical results obtained from the derived formula. We compared our theoretical results with experimental results reported in the literature and the comparison shows a good match. Finally, we performed a comparison between depleted and nondepleted pump regime. The results show a difference of 20% between two efficiencies making nondepleted efficiency equation, which used in many theoretical – experimental comparisons, not accurate at high power densities.

Impact of Artificial Compressibility on the Numerical Solution of Incompressible Nanofluid Flow

The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in previous studies. Therefore, this paper assesses the effect of this factor on the convergence speed and accuracy of results for various types of thermo-flow. To improve the stability and convergence speed of time discretizations, the fifth-order Runge-Kutta method is applied. A computer program has been written in FORTRAN to solve the discretized equations in different Reynolds and Grashof numbers for various grids. The results demonstrate that the artificial compressibility factor has a noticeable effect on the accuracy and convergence rate of the simulation. The optimum artificial compressibility is found to be between 1 and 5. These findings can be utilized to enhance the performance of commercial numerical simulation tools, including ANSYS and COMSOL.