Standard Runge-Kutta Method Research Articles

Smooth Transition from Shakura-Sunyaev Disc toAdvection-Dominated Accretion Flow

We solve a set of basic equations describing black hole accretion flowsusing the standard Runge-Kutta method and a bridging formula for theradiative cooling, and show that a smooth transition from aShakura-Sunyaev disc to an advection-dominated accretion flow isrealizable for the high-viscosity case, without the need of involvingany extra mechanism of energy transport.

A Non-Standard Finite-Difference Scheme for a Model of HIV Transmission and Control

An implicitly-derived explicit non-standard finite-difference scheme will be constructed and used to solve a deterministic non-linear model for HIV transmission and control. The model monitors the populations of untreated susceptibles, vaccinated susceptibles (given prophylactic vaccines), untreated infecteds and treated infecteds as time (t) tends to infinity. Unlike the standard fourth-order Runge-Kutta method (RK4), which induces contrived numerical instabilities, the non-standard method will be seen to be free of numerical instabilities for all parameter values used in the numerical simulations.

Free evolution of self-gravitating, spherically symmetric waves

We perform a numerical free evolution of a self-gravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in the Eddington-Finkelstein coordinates. The simplicity of the system allows us to display and deal with the typical gauge instability present in these coordinates. The numerical evolution is performed with a standard method of lines, fourth order in space and time. The evolution is performed using the standard Runge-Kutta method while the space discrete derivative is symmetric (nondissipative). The constraints are preserved, within numerical errors, by the evolution and we are able to reproduce several known results.

FORCED RESPONSE OF STRUCTURAL DYNAMIC SYSTEMS WITH LOCAL TIME-DEPENDENT STIFFNESSES

This paper deals with a method which is meant to directly approximate the steady state response of linear differential equations with periodic coefficients under external excitations. The interest lies in the use of particular systems with time-independent characteristics (mass, damping) and with periodically time-varying stiffness. A description of the principle of the method is provided. This method has been successfully tested on a single-degree-of-freedom (s.d.o.f) example and compared to the standard Runge–Kutta method. Moreover, the parameters are assumed to be a modification of initial non-parametric systems and allow us the use of the forced reanalysis methods to improve the direct spectral method (DSM). The description of the reanalysis method is made with its implementation within the direct spectral method. Then, a practical application concerning a clamped/free beam with parametric mounts is presented to demonstrate the ability of the proposed method in the analysis of systems which have many d.o.f.s and localized parameters.

Numerical analysis of dynamical systems and the fractal dimension of boundaries

A set of MapleV R.4/5 software routines for calculating the numerical evolution of dynamical systems and flexible plotting the results is presented. The package consists of an initial condition generator (on which the user can impose quite general constraints), a numerical solving manager, plotting commands that allow the user to locate and focus in on regions of possible interest and, finally, a set of routines that calculate the fractal dimension of the boundaries of those regions. A special feature of the software routines presented here is an optional interface in C, permitting fast numerical integration using standard Runge—Kutta methods, or variations, for high precision numerical integration.

A simple and efficient numerical method to study propagation characteristics of nonlinear optical waveguides

A simple numerical method has been developed to study the propagation characteristics of a nonlinear optical waveguide. The method is applicable to planar waveguides as well as fibers. In this method, second-order differential equation is converted into first-order differential equation which is solved by standard fourth-order Runge-Kutta method. Numerical results for nonlinear planar waveguide and optical fibers with power law profiles and Kerr-like nonlinearity have been presented. The results for planar waveguides match very well with the exact results. Present method can be applied to arbitrary refractive index profiles with any kind of nonlinearity.

Consistent Approximations for Optimal Control Problems Based on Runge–Kutta Integration

This paper explores the use of Runge–Kutta integration methods in the construction of families of finite-dimensional, consistent approximations to nonsmooth, control and state constrained optimal control problems. Consistency is defined in terms of epiconvergence of the approximating problems and hypoconvergence of their optimality functions. A significant consequence of this concept of consistency is that stationary points and global solutions of the approximating discrete-time optimal control problems can only converge to stationary points and global solutions of the original optimal control problem. The construction of consistent approximations requires the introduction of appropriate finite-dimensional subspaces of the space of controls and the extension of the standard Runge–Kutta methods to piecewise-continuous functions. It is shown that in solving discrete-time optimal control problems that result from Runge–Kutta integration, a non-Euclidean inner product and norm must be used on the control space to avoid potentially serious ill-conditioning effects.

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods withq≤1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist.

Abnormal convergence of numerical solutions and Runge-Kutta-type methods admitting no such phonemenon

When the standard Runge-Kutta method is applied to a certain system of linear differential equations, the numerical solution converges to the origin but the true solution diverges to infinity. This phenomenon is named the abnormal convergence of numerical solutions. Abnormal convergence has not been studied fully; however, many concepts involving A-stability have been established to treat the phenomenon that the numerical solution diverges even though the true one converges, as is apt to occur in integration of a stiff system. An example will be illustrated here in which the standard Runge-Kutta method generates abnormal convergence. Further, it will be proved that the abnormality is caused by the existence of an intersection of the absolute stability region and the right-half plane. Even if the 1/8 formula method is used instead of the standard one, the same phenomenon can be seen because all four-stage, fourth-order, explicit Runge-Kutta methods have the unique absolute stability region. This paper seeks for Runge-Kutta methods that are A-stable and never generate abnormal convergence, even though this might be an excessive requirement. Concretely, two types of methods will be proposed whose absolute stability region coincides with the left-half plane. Furthermore, it will be reported that Gauss-Legendre methods are typical ones.

On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.

A lumped study of laminar forced convection in pipes with circular transverse fins at the outer surface

A simplified analysis of laminar forced convection heat transfer in an externally finned horizontal pipe was performed using a lumped energy balance formulation. Local values for both the internal and external Nusselt numbers via correlations have been employed, leading to a simple ordinary differential equation which is solved numerically by a standard Runge-Kutta method. Results are presented for the distribution of the dimensionless total heat transferQT/Qmax plotted against the dimensionless axial stationX in the region of thermal development of the pipe. A sample of typical cases covering a wide variety of combinations of geometrical and thermal parameters in the finned pipe are discussed at length. It is found that results for all cases tested compared favorably with other more laborious numerically and analytically-oriented papers published recently in the open literature.

The evolution of nonlinear waves in active media

An evolution equation that describes the behaviour of single waves in active media is derived. The basic mathematical model corresponds to pulse transmission in nerve fibers according to the hyperbolic telegraph equation. A numerical experiment is carried out in which the evolution equation is solved by the pseudospectral method and the corresponding stationary wave equation by the standard Runge-Kutta method. The evolution equation has a stationary solution in the form of an unsymmetric solitary wave with a refractive tail. The numerical simulation of the process gives physically admissible results, namely, the suitable wave profile and the existence of the threshold and asymptotic values. The situation analyzed here in an active medium with energy influx is in a certain sense similar to the formation of solitary waves in a conservative medium.

The Efficient Numerical Calculation of Condensational Cloud Drop Growth

A modified Runge-Kutta integration technique is applied to condensational cloud droplet growth calculations, using the device of limiting dr, the growth of a droplet of radius r over the time step dt to some function of the absolute value of r(eq)-r, where r(eq) denotes the equilibrium value. The use of variable time steps to obtain an efficient choice of dt is also discussed. Results using these techniques are compared with those obtained with the standard fourth-order Runge-Kutta method, showing that the modified technique reduces the requisite computer time by a factor of about one hundred.

Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. I. The nonstiff case

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of variable order block implicit formulae suitable for stiff problems. The central idea is similar to one due to C. W. Gear in developing Runge-Kutta starters for linear multistep methods. Some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard Runge-Kutta methods are made.