Classical Fourth-order Runge-Kutta Method Research Articles

An effective variational iteration algorithm for solving Riccati differential equations

The piecewise variational iteration method (VIM) for solving Riccati differential equations (RDEs) provides a solution as a sequence of iterates. Therefore, its application to RDEs leads to the calculation of terms that are not needed and more time is consumed in repeated calculations for series solutions. In order to overcome these shortcomings, we propose an easy-to-use piecewise-truncated VIM algorithm for solving the RDEs. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the classical fourth-order Runge–Kutta method (RK4) verify that the new method is very effective and convenient for solving Riccati differential equations.

A BEM/FEM Coupling Approach for Fluid-Structure Interaction Simulation of Cell Motion

In this paper, accurate and efficient simulation of cell motion in a biological fluid flow is investigated. The membrane of a moving cell is represented by a thin shell composed of incompressible neo-Hookean elastic materials and the liquids around the membrane are approximated as incompressible Newtonian flows with low Reynolds numbers. The biofluid mechanics is approximated by the Stokes flow equations. A low-order BEM model is developed for the two biological fluids coupled at the membrane surface. The moving boundary problem in fluid mechanics can be effectively solved using the BEM with a GMRES solver. The FEM model based on a flat thin shell element is further developed to predict the membrane load due to the large deformation of a moving cell. Computational efficiency is greatly improved due to the one-dimensional reduction in the present BEM and FEM models. The BEM solver for the biological fluids is coupled with the FEM solver for the cell membrane at the membrane surface. The position of themembrane surface nodes is advanced in time by using the classical fourth-order Runge-Kutta method. Numerical instability is avoided by using a relatively small time step. Further numerical instabilities in the FEM solver is alleviated by using various techniques. The present method is applied to the FSI problems of cell motion in a cylindrical flow. Numerical examples can illustrate the distinct accuracy, efficiency and robustness of the present method. Furthermore, the importance of bending stiffness of a cell membrane for stable cell motion simulation is emphasized. It is suggested that the present approach be an appealing alternative for simulating the fluid-structure interaction of moving cells. AMS subject classifications: 74K25, 74S15, 74S05, 74F10 PACS: 02.70.Dh, 02.30.Rz, 46.70.De, 83.10.-y

A numeric–analytic method for approximating the chaotic Chen system

The epitome of this paper centers on the application of the differential transformation method (DTM) the renowned Chen system which is described as a three-dimensional system of ODEs with quadratic nonlinearities. Numerical comparisons are made between the DTM and the classical fourth-order Runge–Kutta method (RK4). Our work showcases the precision of the DTM as the Chen system transforms from a non-chaotic system to a chaotic one. Since the Lyapunov exponent for this system is much higher compared to other chaotic systems, we shall highlight the difficulties of the simulations with respect to its accuracy. We wrap up our investigations to reveal that this direct symbolic–numeric scheme is effective and accurate.

Development of an improved CNC interpolator and performance evaluation in terms of chordal error and feedrate deviation

The parametric interpolators of modern CNC machines use Taylor's series approximation to generate successive parameter values which, after substituting into the curve equation, gives the x, y, z coordinates of the tool positions. In order to achieve greater accuracy, higher order derivatives are required which complicates the calculation when the curve is represented by NURBS curve. This method calculates the chordal error on a given segment by estimating the curvature which neglects a fraction of the error. In order to avoid calculating higher derivatives and make the calculations easier the classical fourth-order Runge-Kutta (RK) method is proposed in this research, which only requires the first derivative to be calculated, but achieves the accuracy of Taylor's approximation with higher order terms. This paper also proposes the estimation of chordal error on the average value of the parameters at the end points of a given curve segment, which does not require calculation of curvature at every segment. Finally computer simulation is performed on different types of spline curves to show that the proposed method results in reduced chordal error and less fluctuation in feedrate.

Numerical Solutions for Ito Coupled System

In this paper, the classical fourth-order Runge-Kutta method and Heun’s method are applied to initial value problems for system of ordinary differential equations in nonlinear cases which we reach it by Painleve analysis, focusing our interest in Ito coupled nonlinear partial differential system. The equations are solved by scheme of one step methods. Numerical results for the velocity in three dimensions are obtained and reported graphically for various temperatures to show interesting aspects of the solution.

Modelling the attenuation of laminar fluid transients in piping systems

The damping of laminar fluid transients in piping systems is studied numerically using a two-dimensional water hammer model. The numerical scheme is based on the classical fourth order Runge–Kutta method for time integration and central difference expressions for the spatial terms. The results of the present method show that the damping of transients in piping systems is governed by a non-dimensional parameter representing the ratio of the Joukowsky pressure force to the viscous force. In terms of time scales, this non-dimensional parameter represents the ratio of the viscous diffusion time scale to the pipe period. For small values of this parameter, the damping of the fluid transient becomes more pronounced while for large values, the fluid transient is subjected to insignificant damping. Moreover, the non-dimensional parameter is shown to influence other important transient phenomena such as line packing, instantaneous wall shear stress values and the Richardson annular effect.

Simulation electro-optic switch of plasma-electrode Pockels cells driven by one-pulse process

In this paper, a fluid model of electro-optic switch of plasma-electrode Pockels cells (PEPC) driven by one-pulse process is proposed. The model includes a simplified set of fluid continuity equations, Poisson's equation and the electron average energy equation, and these equations are numerically solved with the exponential difference scheme, the classical successive over relaxation method (SOR) and the classical fourth-order Runge-Kutta method, respectively. The evolving characteristics of density of charged particles, electronic temperature, electric field, discharge current, charging voltage of the KDP crystal and the switching efficiency of PEPC are simulated. The model for PEPC, which reveals the relationship between the microcosmic physical processes of gaseous discharge plasma and macrocosmic parameters of PEPC, is very useful for the design of large aperture electro-optic switches.

Effective numerical algorithm for fiber amplifiers

Based on the Dormand-Prince (DP) formula, a fast and effective method is proposed, which can implement automatic step-size adjustment to solve the model equations of fiber Raman amplifiers and erbium-doped fiber amplifiers (EDFAs). Numerical results show that (1) the CPU time of the proposed method is about 4.2 times smaller than that of the DP method when the Raman amplifier propagation equations are solved; (2) our method reduces the running time by more than two orders of magnitude in comparison with the classical fourth-order Runge-Kutta method; (3) in comparison with the DP method, the proposed method can increase the computational speed by about 2.5 times in solving the model equations of EDFAs.

Effective methods for Raman amplifier propagation equations

Fast and accurate methods for Raman amplifier propagation equations are proposed. Numerical results demonstrate that these methods can increase the accuracy by more than one or two orders of magnitude in comparison with the classical methods, and the computing speed can increase more than 16 times compared to the classical fourth-order Runge-Kutta method on the same conditions.

A Composite Runge–Kutta Method for the Spectral Solution of Semilinear PDEs

A new composite Runge–Kutta (RK) method is proposed for semilinear partial differential equations such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev–Petviashvili (KP), Kuramoto–Sivashinsky (KS), Cahn–Hilliard, and others having high-order derivatives in the linear term. The method uses Fourier collocation and the classical fourth-order RK method, except for the stiff linear modes, which are treated with a linearly implicit RK method. The composite RK method is simple to implement, indifferent to the distinction between dispersive and dissipative problems, and as efficient on test problems for KS and KP as any other generally applicable method.

Method of reduction of order for solving singularly perturbed two-point boundary value problems

In this paper, a method of reduction of order is proposed for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. It is distinguished by the following fact: the original singularly perturbed boundary value problem is replaced by a pair of initial value problems. Classical fourth order Runge–Kutta method is used to solve these initial value problems. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory.

Runge–Kutta methods for linear ordinary differential equations

Three new Runge–Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODEs) with constant coefficients. Such ODEs arise in the numerical solution of partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge–Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge–Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge–Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

SOME MODIFIED RUNGE–KUTTA METHODS FOR THE NUMERICAL SOLUTION OF SOME SPECIFIC SCHRÖDINGER EQUATIONS AND RELATED PROBLEMS

Some new modified Runge–Kutta methods with minimal phase lag are developed for the numerical solution of the eigenvalue Schrödinger equation and related problems with oscillating solutions. These methods are based on the very well-known Runge–Kutta method of order 4. For the numerical solution of the eigenvalue Schrödinger equation, we investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x; it is assumed, also, that the wave functions tend to zero for x → ±∞; (ii) the general case for the well-known cases of the Morse potential and Woods–Saxon or optical potential. Also, we have applied the new methods to some well-known problems with oscillatory solutions. Numerical and theoretical results show that this new approach is more efficient than the well-known classical fourth order Runge–Kutta method and the Numerov method.

A fourth order scheme for nonlinear integral equations of viscoelasticity

A modification of the classical fourth-order Runge-Kutta method has been developed to solve certain nonlinear integral equations that arise in modeling stress-temperature relationships in viscoelastic films. Here we present the numerical scheme and apply it to the calculation of temperature paths which minimize residual stress in the material.

Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials

In this paper we consider three examples of discontinuous Sturm-Liouville problems with symmetric potentials. The eigenvalues of the systems were determined using the classical fourth order Runge-Kutta method. These eigenvalues are used to reconstruct the potential function using an algorithm presented in Kobayashi [1, 2]. The results of our numerical experiments are discussed.

Nonlinear response of elastic plates to pulse excitations

The dynamic analogue of the Von Karman's equation is used to study the nonlinear response of elastic plates of square and circular geometry with simply supported and clamped-in boundary conditions and immovably constrained and stress free edge conditions, subjected to exponentially decaying (used for an adequate description of a blast load) cosine and exponential asymptotic step pulse excitations. Transformation of the dependent time function such that the solution of a linear system subjected to the same pulse function is used as an additional transforming function brings the time differential equation into a form in which the ultraspherical polynomial approximation technique can be applied to get the nonlinear response, which is compared with the digital solution obtained on the WIPRO:B-200 computing system by using the classical fourth-order Runge-Kutta method.

Discrete methods for boundary value problems with applications in plate deflection theory

In this report we survey numerical techniques of order 2, 4 and 6 for the solution of a two-point boundary value problem associated with a fourth-order linear ordinary differential equation. A sufficient condition guaranteeing a unique solution of the boundary value problem is also given. Numerical results are tabulated for two typical numerical examples and compared with some known methods including the shooting technique employing the classical fourth-order Runge-Kutta method.

Rotational Temperature in an Underexpanded Jet

The axial rotational temperature distribution in an underexpanded nitrogen jet emitted from a sonic orifice has been investigated. The expansion is from room temperature and the gas possesses negligible vibrational energy. The observed difference between the rotational temperature measured with an electronbeam probe and the isentropic value can be described by a rotational relaxation analysis. The governing flow equations were integrated numerically by the classical fourth-order Runge-Kutta method. The rotational relaxation time developed by Lordi using Wang Chang and Uhlenbeck's formal kinetic theory and Parker's molecular model was employed. Both the easy and difficult energy transfer cases were investigated and the easy case was shown to underestimate nonequilibrium affects. The numerical solutions were very sensitive to the separation between the repulsive force centers, for both energy transfer cases. The rotational temperature distributions obtained in this study are in general agreement with existing experimental data for nitrogen.