Classical Fourth-order Runge-Kutta Research Articles

A comparison of classical Runge-Kutta and Henon’s methods for capturing chaos and chaotic transients in an aeroelastic system with freeplay nonlinearity

A typical two-dimensional airfoil with freeplay nonlinearity in pitch undergoing subsonic flow is studied via numerical integration methods. Due to the existence of the discontinuous nonlinearity, the classical fourth-order Runge-Kutta (RK4) method cannot capture the aeroelastic response accurately. Particularly, it is because the RK4 method is incapable of detecting the discontinuous points of the freeplay that leads to the numerical instability and inaccuracy. To resolve this problem, the RK4 method is used with the aid of the Henon’s method (referred to as the RK4Henon method) to precisely predict the freeplay’s switching points. The comparison of the classical RK4 and the RK4Henon methods is carried out in the analyses of periodic motions, chaos, and long-lived chaotic transients. Numerical simulations demonstrate the advantages of the RK4Henon method over the classical RK4 method, especially for the analyses of chaos and chaotic transients. Another existing method to deal with the freeplay nonlinearity is to use an appropriate rational polynomial (RP) to approximate this discontinuous nonlinearity. Consequently, the discontinuity is removed. However, it is demonstrated that the RP approximation method is unable to capture the chaotic transients. In addition, an efficient tool for predicting the existence of chaotic transients is proposed by means of the evolution curve of the largest Lyapunov exponent. Finally, the effects of system parameters on the aeroelastic response are investigated.

Simulation of forced deformable bodies interacting with two-dimensional incompressible flows: Application to fish-like swimming

We present an efficient algorithm for simulation of deformable bodies interacting with two-dimensional incompressible fluid flows. The temporal and spatial discretizations of the Navier–Stokes equations in vorticity stream-function formulation are based on classical fourth-order Runge–Kutta scheme and compact finite differences, respectively. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero over an optimal grid. For introducing a deformable body in fluid flow, the volume penalization method is used. A Lagrangian structured grid with prescribed motion covers the deformable body which is interacting with the surrounding fluid due to the hydrodynamic forces and the torque calculated on the Eulerian reference grid. An efficient law for controlling the curvature of an anguilliform fish, swimming toward a prescribed goal, is proposed which is based on the geometrically exact theory of nonlinear beams and quaternions. Validation of the developed method shows the efficiency and expected accuracy of the algorithm for fish-like swimming and also for a variety of fluid/solid interaction problems.

A Fast Immersed Boundary Fourier Pseudo-spectral Method for Simulation of the Incompressible Flows

Abstract The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier-Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to some other similar methods, there is not an explicit external forcing function in the present formulation. At the beginning of each time step, the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The classical explicit fourth-order Runge-Kutta method is used in time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increase the time accuracy. The method is employed in simulation of some different test cases, including the flow behind impulsively started circular cylinder, oscillating circular cylinder in fluid at rest and insect-like flapping wing motion. The results show accuracy and efficiency of the method.

Effects of parallel strategies in the transitional flow investigation

The direct numerical simulation of transitional and turbulent incompressible flows is an area that is increasing with the advance in computational resources. Code parallelization has became a useful tool in these simulations. In the present paper the unsteady two dimensional Navier–Stokes equations are used as physical model. Tollmien–Schlichting waves propagating in a Poiseuille flow is adopted as test case. Three frequencies showing different behavior according to the linear stability theory (LST) are used: unstable, nearly neutral and stable waves. The parallelization is done via domain decomposition in the streamwise direction. The time derivative is integrated by a classical fourth-order Runge–Kutta method. High-order compact finite difference schemes are used for the spatial derivatives discretization. The Poisson equation is solved by multigrid methods. The present paper explores different parallelization techniques for solving the tridiagonal system, and the multigrid method. The results are compared with LST, and also between the different parallel strategies to show the advantages and disadvantages of each one.

Wavelet solutions of Burgers’ equation with high Reynolds numbers

A wavelet method is proposed to solve the Burgers’ equation. Following this method, this nonlinear partial differential equation is first transformed into a system of ordinary differential equations using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. Such a wavelet-based solution procedure has been justified by solving two test examples: results demonstrate that the proposed method has a much better accuracy and efficiency than many other existing numerical methods, and whose order of convergence can go up to 5. Most importantly, our results also indicate that the present wavelet method can readily deal with those fluid dynamics problems with high Reynolds numbers.

Sin-Cos-Taylor-Like method for solving stiff ordinary diffrential equations

This paper discusses the derivation of an explicit Sin-Cos-Taylor-Like method for solving stiff ordinary differential equations, which is a formulation of the combination of a polynomial and the exponential function. This new method requires extra work to evaluate a number of differentiations of the function involved. However, the result shows smaller errors when compared to the results from the explicit classical fourth-order Runge-Kutta (RK4) and the Adam-Bashforth-Moulton (ABM) methods. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. Although extra work is required, this explicit method has its own advantages. Besides providing excellent results, the cost of computation using this explicit method is much cheaper than the implicit methods. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge-Kutta method is not adequate for the solution of stiff problems. As a result, we find that this explicit method is of order-6, which has been developed, and proved to be both A-stable and L-stable.

Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh

In this study, the method of lines (MOL) has been applied to solve two-dimensional vertically integrated shallow water equations in Cartesian coordinates for the prediction of water levels due to a storm surge along the coast of Bangladesh. In doing so, the partial derivatives with respect to the space variables were discretized by the finite difference (central) method to obtain a system of ordinary differential equations (ODEs) with time as independent variable. The classical fourth-order Runge–Kutta method was used to solve the obtained system of the ODEs. We used a nested finite difference scheme, where a high resolution fine grid model (FGM) capable of incorporating all major islands along the coastal region of Bangladesh was nested into a coarse grid model (CGM) covering up to 15°N latitude of the Bay of Bengal. The boundaries of the coast and islands were approximated through proper stair step. Appropriate tidal condition over the model domain was generated by forcing the sea level to be oscillatory with the constituent M 2 along the southern open boundary of the CGM omitting wind stress. Along the northeast corner of the FGM, the Meghna River discharge was taken into account. The developed model was applied to estimate water levels along the coast of Bangladesh due to the interaction of tide and surge associated with the April 1991 storm. We also computed our results employing the standard finite difference method (FDM). Simulated results show the MOL performs well in comparison with the FDM with regard to CPU time and stability, and ensures conformity with observations.

The multi-step homotopy analysis method for modified epidemiological model for computer viruses

In this paper, we consider the modified epidemiological model for computer viruses (SAIR) proposed by Piqueira and Araujo (Appl Math Comput 2(213):355–360, 2009). The multi-step homotopy analysis method (MHAM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MHAM and the classical fourth-order Runge-Kutta method reveal that this method is very effective. The solutions obtained are also presented graphically.

Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameterhby following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

Runge–Kutta methods for a semi-analytical prediction of milling stability

On the basis of Runge–Kutta methods, this paper proposes two semi-analytical methods to predict the stability of milling processes taking a regenerative effect into account. The corresponding dynamics model is concluded as a coefficient-varying periodic differential equation with a single time delay. Floquet theory is adopted to predict the stability of machining operations by judging the eigenvalues of the state transition matrix. This paper firstly presents the classical fourth-order Runge–Kutta method (CRKM) to solve the differential equation. Through numerical tests and analysis, the convergence rate and the approximation order of the CRKM is not as high as expected, and only small discrete time step size could ensure high computation accuracy. In order to improve the performance of the CRKM, this paper then presents a generalized form of the Runge–Kutta method (GRKM) based on the Volterra integral equation of the second kind. The GRKM has higher convergence rate and computation accuracy, validated by comparisons with the semi-discretization method, etc. Stability lobes of a single degree of freedom (DOF) milling model and a two DOF milling model with the GRKM are provided in this paper.

The multistage homotopy analysis method: application to a biochemical reaction model of fractional order

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order.

A Wavelet Method for Solving Nonlinear Time-Dependent Partial Differential Equations

A wavelet method is proposed for solving a class of nonlinear time-dependent partial differential equations. Following this method, the nonlinear equations are first transfoulled into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers' equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and the order of convergence of such a wavelet method can even reach about 5.

Perfectly Matched Layer as an Absorbing Boundary Condition for Computational Aero-Acoustic

Advances in Computational Aeroacoustics (CAA) depend critically on the availability of accurate, nondispersive, least dissipative computation algorithm as well as high quality numerical boundary treatments. This paper focuses on the Perfectly Matched Layer (PML) for external boundaries in CAA. To achieve low dissipation and dispersion errors, Dispersion-Relation-Preserving (DRP) Schemes are used for spatial discretization of the acoustic equations. The classical fourth-order Runge-Kutta time scheme is applied to the acoustic equations for time discretization. Four cases are given to illustrate the 2D PML equations for the linearized/nonlinear Euler equations in Cartesian coordinates and Cylindrical coordinates. The results show that the PML is effective as absorbing boundary condition. Those are basis for PML in actual computations of acoustic problems.

Fourier Spectral Methods for Solving Some Nonlinear Partial Differential Equations

The spectral collocation or pseudospectral (PS) methods (Fourier transform methods) combined with temporal discretization techniques to numerically compute solutions of some partial differential equations (PDEs). In this paper, we solve the Korteweg-de Vries (KdV) equation using a Fourier spectral collocation method to discretize the space variable, leap frog and classical fourth-order Runge-Kutta scheme (RK4) for time dependence. Also, Boussinesq equation is solving by a Fourier spectral collocation method to discretize the space variable, finite difference and classical fourth-order RungeKutta scheme (RK4) for time dependence. Our implementation employs the Fast Fourier Transform (FFT) algorithm.

Direct numerical methods dedicated to second-order ordinary differential equations

This article presents numerical methods for solving second-order ordinary differential equations. These methods are based on Hermite polynomials, which makes them more computationally effective than, for example, the classical fourth-order Runge–Kutta method. In addition, the presented algorithms were modified to reduce the CPU time required. Hermite polynomials are not very sensitive to the Runge phenomenon; moreover, the numerical errors of interpolation are relatively small for large time steps, which is an advantage. These methods are presented in the form of pseudo-code for easier application. The presented approach to numerical methods is a result of simulated, strongly non-linear vibrations with contact phenomena such as Coulomb friction and impact.

Comparative study of travelling-wave and numerical solutions for the coupled short pulse (CSP) equation

The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge—Kutta scheme.

A New Method for Solving Singularly Perturbed Boundary Value Problems

In this paper, a new initial value method for solving a class of nonlinear singularly perturbed boundary value problems with a boundary layer at one end is proposed. The method is designed for the practicing engineer or applied mathematician who needs a practical tool for these problems (easy to use, modest problem preparation and ready computer implementation). Using singular perturbation analysis the method is distinguished by the following fact: the original problem is replaced by a pair of first order initial value problems; namely, a reduced problem and a boundary layer correction problem. These initial value problems are solved using classical fourth order Runge-Kutta method. Numerical examples are given to illustrate the method. It is observed that the present method approximates the exact solution very well.

AN INITIAL VALUE TECHNIQUE FOR SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS WITH A SMALL NEGATIVE SHIFT

Abstract. In this paper, we present an initial value technique for solvingsingularly perturbed differential difference equations with a boundary layerat one end point. Taylor’s series is used to tackle the terms containing shiftprovided the shift is of small order of singular perturbation parameter andobtained a singularly perturbed boundary value problem. This singularlyperturbed boundary value problem is replaced by a pair of initial valueproblems. Classical fourth order Runge–Kutta method is used to solvethese initial value problems. The effect of small shift on the boundarylayer solution in both the cases, i.e., the boundary layer on the left sideas well as the right side is discussed by considering numerical experiments.Several numerical examples are solved to demonstate the applicability ofthe method.AMS Mathematics Subject Classification : 65L11.Key words and phrases : Initial value technique, singular perturbation,differential–difference equation, boundary layer. 1. IntroductionThe numerical treatment of singular perturbations is far from the trivial be-cause of the boundary layer behavior. In recent decades this is a field of increas-ing interest to applied mathematicians and numerical analysts in view of thechallenges the problems there in pose to the researchers. For a detailed theoryand analytical discussion on singular perturbation problems one may refer tothe books and high level monographs: O’ Malley [13], Nayfeh [12], Kevorkianand Cole [11], Bender and Orszag [10]. A Singularly perturbed delay differentialequation is an ordinary differential equation in which the highest derivative ismultiplied by a small parameter and involving atleast one delay term. In 1979,the Japanese physicist Kensuke Ikeda considered a nonlinear absorbing medium

Effect of Gravity Modulation on Weakly Non-Linear Stability of Stationary Convection in a Dielectric Liquid

The effect of time-periodic oscillations of the RayleighBenard system on the heat transport in dielectric liquids is investigated by weakly nonlinear analysis. We focus on stationary convection using the slow time scale and arrive at the real GinzburgLandau equation. Classical fourth order Runge-kutta method is used to solve the Ginzburg-Landau equation which gives the amplitude of convection and this helps in quantifying the heat transfer in dielectric liquids in terms of the Nusselt number. The effect of electrical Rayleigh number and the amplitude of modulation on heat transport is studied. Keywords—Dielectric liquid; Nusselt number; Amplitude equation.