RK4 Method Research Articles

An Explicit Fourth-Order Runge–Kutta Method for Dynamic Force Identification

This paper presents a new technique for input reconstruction based on the explicit fourth-order Runge–Kutta (RK4) method. First, the state-space representation of the dynamic system is discretized by the explicit RK4 method under the assumption of linear interpolation for the dynamic load, leading to a recurrence equation between the current state and the previous state. Then, the mapping from the sequences of input to output is established through the recursive operation of the system equation and observation equation. Finally, the stabilized force information is recovered using the Tikhonov regularization method. This approach makes use of the good stability and high precision of the RK4 method; in addition, the computational efficiency is enhanced by avoiding the computation of the inverse stiffness matrix. The proposed method is numerically illustrated and validated with various excitations on a simple four-story shear building and a more complicated 2D truss structure, along with a detailed parametric study. The simulation studies show that the external loads can be reconstructed with high efficiency and accuracy under a low noise environment.

A variational numerical method based on finite elements for the nonlinear solution characteristics of the periodically forced Chen system

Nonlinear dynamical systems and their solutions are very sensitive to initial conditions and therefore need to be approximated carefully. In this article, we present and analyze nonlinear solution characteristics of the periodically forced Chen system with the application of a variational method based on the concept of finite time-elements. Our approach is based on the discretization of physical time space into finite elements where each time-element is mapped to a natural time space. The solution of the system is then determined in natural time space using a set of suitable basis functions. The numerical algorithm is presented and implemented to compute and analyze nonlinear behavior at different time-step sizes. The obtained results show an excellent agreement with the classical RK-4 and RK-5 methods. The accuracy and convergence of the method is shown by comparing numerically computed results with the exact solution for a test problem. The presented method has shown a great potential in dealing with the solutions of nonlinear dynamical systems and thus can be utilized in delineating different features and characteristics of their solutions.

Outcomes of Non-uniform Heat Source/Sink on Micropolar Nanofluid Flow in Presence of Slip Boundary Conditions

Present study communicates the theoretical outline for the flow of an incompressible electrically conducting micropolar nanofluid over a moving vertical surface in accordance with surface skid and jump of temperature in attendance of non-consistent heat source/sink. The leading flow connected equations are condensed to a collection of coupled ODEs by introducing similarity transformations and then unravelled numerically by means of RK-4 method with shooting practice. The obtained results are presented through diagram, tables and the physical sides of the problem are talked about.

Dynamic analysis of an aeroelastic airfoil with freeplay nonlinearity by precise integration method based on Padé approximation

An improved precise integration method (PIM) incorporated with Pade approximation (PadePIM) is proposed, and the aeroelastic behavior of an aeroelastic airfoil with freeplay nonlinearity is investigated by using the PadePIM method. For the aeroelastic system with this non-smooth nonlinearity, a predictor-correction algorithm is adopted to avoid the numerical inaccuracy induced by the crossover of the switching points of freeplay nonlinearity. The comparative studies for the PadePIM method and other conventional numerical methods, such as the classical fourth-order Runge–Kutta (RK4) method, the RK4 method combined with the Henon’s method and the original PIM, which is based on the Taylor series expansion, are performed for the solution of dynamic responses of the aeroelastic airfoil. Numerical results verify that the PadePIM method is unconditionally stable and has higher accuracy and efficiency than other methods, especially with high reliability for obtaining complex dynamic responses. When freeplays in pitch/plunge degrees-of-freedom are further considered for nonlinear aeroelastic characteristics of the aeroelastic airfoil, the regions of limit cycle oscillation and chaotic motions are detected for speeds below the stability boundary of the system predicted by linear theory. Results of the present study show that the existence of these regions is strongly dependent on the freeplay magnitudes, which have significant effects on the aeroelastic responses. These nonlinear aeroelastic behaviors exhibited in the system with multiple freeplay nonlinearities are greatly different from that with single freeplay nonlinearity. It has also been shown that the appropriate combinations of multiple freeplays can be selected for eliminating the complex dynamic responses of the aeroelastic airfoil.

Bayesian inference for higher-order ordinary differential equation models

Often the regression function appearing in fields like economics, engineering, and biomedical sciences obeys a system of higher-order ordinary differential equations (ODEs). The equations are usually not analytically solvable. We are interested in inferring on the unknown parameters appearing in such equations. Parameter estimation in first-order ODE models has been well investigated. Bhaumik and Ghosal (2015) considered a two-step Bayesian approach by putting a finite random series prior on the regression function using a B-spline basis. The posterior distribution of the parameter vector is induced from that of the regression function. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. Bhaumik and Ghosal (2016) remedied this by directly considering the distance between the function in the nonparametric model and a Runge–Kutta (RK4) approximate solution of the ODE while inducing the posterior distribution on the parameter. They also studied the convergence properties of the Bayesian method based on the approximate likelihood obtained by the RK4 method. In this paper, we extend these ideas to the higher-order ODE model and establish Bernstein–von Mises theorems for the posterior distribution of the parameter vector for each method with n − 1 / 2 contraction rate.

Influence of Variable Fluid Properties on Nanofluid Flow over a Wedge with Surface Slip

The present article explores the influence of variable fluid properties on nanofluid flow over a wedge. The flow analysis has been considered under the effect of surface slip. It is supposed to have fluid viscosity and thermal conductivity as an inverse function and linear function of temperature, respectively. The resulting nonlinear ordinary differential equations are solved numerically using RK-4 method together with shooting procedure. The consequence of involved pertinent parameters on the flow province has been discussed through graphs and tables coupled with required discussion. Our analysis conveys that the fluid temperature is higher in the presence of variable viscosity parameter and thermal conductivity parameter. It is also observed that the wall stress decreases with increasing velocity slip parameter.

Stefan blowing effects on MHD bioconvection flow of a nanofluid in the presence of gyrotactic microorganisms with active and passive nanoparticles flux

The present paper investigates the effect of Stefan blowing on the hydro-magnetic bioconvection of a water-based nanofluid flow containing gyrotactic microorganisms through a permeable surface. Also we studied both actively and passively the controlled flux of nanoparticles and the effect of a surface slip at the wall. We adopt a similarity approach to reduce the leading partial differential equations into ordinary differential equations along with two separate boundary conditions (active and passive) and solve the resulting equations numerically by employing the RK-4 method through the shooting technique to perform the flow analysis. Discussions on the effect of emerging flow parameter on the flow characteristic are made properly through graphs and charts. We observed that the effects of the traditional Lewis number and suction/blowing parameter on temperature distribution and microorganism concentration are converse to each other. A fair result comparison of the present paper with formerly obtained results is given.

Numerical Simulation Using GEM for the Optimization Problem as a System of FDEs

In this paper, we introduce a numerical treatment using generalized Euler method (GEM) for the non-linear programming problem which is governed by a system of fractional differential equations (FDEs). The appeared fractional derivatives in these equations are in the Caputo sense. We compare our numerical solutions with those numerical solutions using RK4 method. The obtained numerical results of the optimization problem model show the simplicity and the efficiency of the proposed scheme.

Numerical study on the resonance response of spar-type floating platform in 2-D surface wave

This paper is concerned with the numerical study on the resonance response of a rigid spar-type floating platform in coupled heave and pitch motion. Spar-type floating platforms, widely used for supporting the offshore structures, offer an economic advantage but those exhibit the dynamically high sensitivity to external excitations due to their shape at the same time. Hence, the investigation of their dynamic responses, particularly at resonance, is prerequisite for the design of spar-type floating platforms which secure the dynamic stability. Spar-type floating platform in 2-D surface wave is assumed to be a rigid body having 2-DOFs, and its coupled dynamic equations are analytically derived using the geometric and kinematic relations. The motion-variance of the metacentric height and the moment of inertia of floating platform are taken into consideration, and the hydrodynamic interaction between the wave and platform motions is reflected into the hydrodynamic force and moment and the frequency-dependent added masses. The coupled nonlinear equations governing the heave and pitch motions are solved by the RK4 method, and the frequency responses are obtained by the digital Fourier transform. Through the numerical experiments to the wave frequency, the resonance responses and the coupling in resonance between heave and pitch motions are investigated in time and frequency domains.

Analysis of photonic crystal transmission properties by the precise integration time domain

Photonic crystals are materials patterned with a periodicity in the dielectric constant, which can create a range of forbidden frequencies called as a photonic band gap. The photonic band gap of the photonic crystal indicates its primary property, which is the basis of its application. In recent years, photonic crystals have been widely used to design optical waveguides, filters, microwave circuits and other functional devices. Therefore, the study on the transmission properties in photonic crystal is significantly important for constructing the optical devices. The finite difference time domain (FDTD) is a very useful numerical simulation technique for solving the transmission properties of the photonic crystals. However, as the FDTD method is based on the second order central difference algorithm, its accuracy is relatively low and the Courant stability condition must be satisfied when this method is used, which may restrict its application. To increase the accuracy and the stability, considerable scientific interest has been attracted to explore the schemes to improve the performance of the FDTD. The fourth order Ronge-Kutta (RK4) method has been applied to the FDTD method, which improves the accuracy and eliminates the influence of accumulation errors of the results, but the stability remains very poor if the time step is large. An effective time domain algorithm based on the high precision integration is proposed to solve the transmission properties of photonic crystals. The Yee cell differential technique is used to discretize the first order Maxwell equations in the spatial domain. Then the discretized Maxwell equations with the absorption boundary conditions and the expression of excitation source are rewritten in the standard form of the first order ordinary differential equation. According to the precise division of the time step and the additional theorem of exponential matrix, the high precision integration is used to obtain the homogeneous solution. To obtain the discretized electric and magnetic fields, the particular solution must be solved based on the excitation and then be added to the homogeneous solution. The transmission properties of photonic crystals are obtained by the Fourier transform. Practical calculation of photonic crystals is carried out by the precise integration time domain, and the accuracy and the stability are compared with those from the FDTD and the RK4 methods. The numerical results show that the precise integration time domain has a higher calculation precision and overcomes the restriction of stability conditions on the time step, which provides an effective analytical method of studying the transmission properties of photonic crystals.

Effect of Operational Parameters on the Production of a Solar Distiller Coupled to a Hybrid Photovoltaic Thermal Collector

It is universally recognized that the lack of water on the one hand, and the depletion of fossil fuels on the other hand are one of the major challenges of our century, the face these critical issues, desalination of salt water and/or brackish water appears as one of the possible solutions to the survival of humanity. Among the techniques used in this field, and needs relatively low drinking water, solar distillation can be a very good solution especially for arid and desert zones. In order to improve the production of solar stills, our work focuses on the coupling of a flat plate solar distiller with a flat collector to ensure preheating distilled water. We prepared heat balances at the distiller and describe their sensor transient thermal behavior. Next, we used the RK4 method to solve systems of equations obtained. The numerical results clearly show the influence of various parameters on the daily production of this system.

Framing the effects of solar radiation on magneto-hydrodynamics bioconvection nanofluid flow in presence of gyrotactic microorganisms

The present article investigates the effects of solar radiation on hydromagnetic bioconvection of water-based nanofluid flow in presence of gyrotactic microorganisms past a permeable surface. The flow analysis has been considered under the effect of surface slip condition. The resulting partial differential equations take the form of ordinary differential equation by employing suitable transformation and then solved numerically using RK-4 method together with shooting technique. A parametric study on the flow characteristics are presented through tables and graphs coupled with required discussion and physical implication. A comparative study has been taken into account between previously published literature and present work to show the efficiency of our investigation and it shows excellent accord.

Mixed convection flow with non-uniform heat source/sink in a doubly stratified magnetonanofluid

In this study, we explore the unsteady flow of viscous nanofluid driven by an inclined stretching sheet. The novelty of the present study is to account for the effect of a non-uniform heat source/sink in a thermally and solutally stratified magnetonanofluid. Governing system of nonlinear partial differential equations is converted into a system of nonlinear ordinary differential equations. Solution of the transformed system is obtained using RK4 method with shooting technique. It is observed that increase in the values of thermal and mass stratification parameter reduce the velocity profile and increase in the values of variable thermal conductivity parameter and non-uniform heat source/sink parameters enhance the temperature distribution. Moreover, skin friction coefficient, Nusselt number and Sherwood number are discussed. Obtained results are displayed both graphically and in tabular form to illustrate the effect of different parameters on the velocity, temperature and concentration profiles. Numerical results are compared with previous published results and found to be in good agreement for special cases of the emerging parameters.

Effect of magnetic field on slip flow of nanofluid induced by a non-linear permeable stretching surface

In this effort the influence of magnetic field on boundary layer nanofluid flow induced by a non-linear permeable stretching sheet has been explored. The flow is happening due to stretching surface along with slip conditions in boundary layer. The governing equations are solved numerically by means of a robust programming MAPLE-17 which includes RK-4 method accompanied by shooting criteria. The impact of appropriate factors on flow are deliberated and physical characteristics of the flow are exhibited through graphs and tables and discussed from the reasonable judgment. Our analysis conveys that the reduced Nusselt number amplifies as the thermophoresis parameter and Brownian motion parameter enhances. On the other hand, it is perceived that thermal boundary layer thickness moderates with the growth of thermal slip parameter. Moreover, it is thought-provoking to find that the concentration distribution declines as the Lewis number escalates. To sum up, the verdict of the current study is that the proficiency of a thermal system can be enriched by scheming the strength of applied magnetic field as well as by electing the applicable values of the physical parameters.

The squeezing flow of Cu-water and Cu-kerosene nanofluids between two parallel plates

The present article investigates the squeezing flow of two types of nanofluids such as Cu-water and Cu-kerosene between two parallel plates in the presence of magnetic field. The governing non-linear partial differential equations are transformed into ordinary differential equations by applying suitable similarity transformation and then solved numerically using RK-4 method with shooting technique and analytically using differential transformation method (DTM). The influence of arising relevant parameters on flow characteristics has been discussed through graphs and tables. A comparative study has been taken into account between existing results and present work and it is found to be in excellent harmony.

Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation

This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically.

Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients

Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

On heat and mass transfer analysis for the flow of a nanofluid between rotating parallel plates

In the present study, we have considered the three dimensional heat and mass transfer with magnetic effects for the flow of a nanofluid between two parallel plates in a rotating system. The reduced governing equations (in a system of ordinary differential equations) are solved by applying Homotopy Analysis Method (HAM). To see the validity of analytical solution, a numerical solution using RK-4 method coupled with shooting method has also been sought. Both the solutions are found to be in excellent agreement. To capture the effects of involved physical parameters graphical representation of the flow are included with comprehensive discussions. It has been observed that the thermophoresis and Brownian motion parameters are directly related to heat transfer but are inversely related to concentration profile. It is also recorded that the higher Coriolis forces decrease the temperature boundary layer thickness.

The Fourth Order Runge-Kutta Spreadsheet Calculator Using VBA Programing for Ordinary Differential Equations

Motivated by the previous literature works of spreadsheet solutions of ordinary differential equations (ODE) and a system of ODEs using fourth-order Runge-Kutta (RK4) method, we have built a spreadsheet calculator for solving ODEs numerically by using the RK4 method and VBA programming. In this spreadsheet calculator, a user interface input form is developed to capture the needed information to solve the ODE via the RK4 method. In the first frame of the user interface input form, users are prompted to enter independent and dependent variables in the ODE. On the other hand, they are needed to input the interval for the independent variables, initial value for the dependent variable, step size h and select the desired accuracy of calculation in the second frame of the user interface input form. Lastly, in the third frame of the user interface input form, users are required to enter the ODE function and exact function of ODE if it exists. Once the Solve button is clicked, the full solution of the ODE will be calculated automatically using the RK4 method. This spreadsheet calculator is user-friendly especially for users who are not familiar with Excel. We hope that this spreadsheet calculator can be used as a marking scheme for educators and students who require the full solutions of the ODEs. Moreover, this spreadsheet calculator reduces calculation time and it is hoped to improve students’ learning ability.

High accuracy solution of bi-directional wave propagation in continuum mechanics

Solution of partial differential equations by numerical method is strongly affected due to numerical errors, which are caused mainly by deviation of numerical dispersion relation from the physical dispersion relation. To quantify and control such errors and obtain high accuracy solutions, we consider a class of problems which involve second derivative of unknowns with respect to time. Here, we analyse numerical metrics such as the numerical group velocity, numerical phase speed and the numerical amplification factor for different methods in solving the model bi-directional wave equation (BDWE). Such equations can be solved directly, for example, by Runge–Kutta–Nyström (RKN) method. Alternatively, the governing equation can be converted to a set of first order in time equations and then using four-stage fourth order Runge–Kutta (RK4) method for time integration. Spatial discretisation considered are the classical second and fourth order central difference schemes, along with Lele's central compact scheme for evaluating second derivatives. In another version, we have used Lele's scheme for evaluating first derivatives twice to obtain the second derivative. As BDWE represents non-dissipative, non-dispersive dynamics, we also consider the canonical problem of linearised rotating shallow water equation (LRSWE) in a new formulation involving second order derivative in time, which represents dispersive waves along with a stationary mode. The computations of LRSWE with RK4 and RKN methods for temporal discretisation and Lele's compact schemes for spatial discretisation are compared with computations performed with RK4 method for time discretisation and staggered compact scheme (SCS) for spatial discretisation by treating it as a set of three equations as reported in Rajpoot et al. (2012) [1].