Runge-Kutta algorithms

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

The Runge-Kutta algorithm cannot handle so-called stiff problems. Computation times are astronomical and thus the algorithm is useless, for that class of ordinary differential equations, specialised stiff solvers have been developed. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps and/or very high and very low concentrations. As a typical example we model an oscillating reaction in The Belousov-Zhabotinsky (BZ) Reaction (p.95). 

This function is called numerous times from the Matlab ODE solver. In the example it is the ode45 which is the standard Runge-Kutta algorithm. ode45 requires as parameters the file name of the inner function, ode autocat. m, the vector of initial concentrations, cO, the rate constants, k, and the total amount of time for which the reaction should be modelled (20 time units in the example). The solver returns the vector t at which the concentrations were calculated and the concentrations themselves, the matrix C. Note that due to the adaptive step size control, the concentrations are computed at times t which are not predefined. 

Niunerical algorithms for solving the GLE are readily available. Only recently, Hershkovitz has developed a fast and efficient 4th order Runge-Kutta algorithm. Memory friction does not present any special problem, especially when expanded in terms of exponentials, since then the GLE can be represented as a finite set of memoiy-less coupled Langevin equations. " Alternatively (see also the next subsection), one can represent the GLE in terms of its Hamiltonian equivalent and use a suitable discretization such that the problem becomes equivalent to that of motion of the reaction coordinate coupled to a finite discrete bath of harmonic oscillators. ... 

Thus, we obtain the second order Runge-Kutta algorithm ... 

This equation is usually integrated using e.g a Runge-Kutta algorithm. The parameters are To, A1A2 and E1+E2 addition to the initial value 0a. 

The Runge-Kutta algorithm is entered into a spreadsheet, and the two Stefan-Maxwell equations (12.196 and 12.197) are integrated using a step size Z/200, with guessed values for the fluxes Ni, N2. The calculated mole fractions of species 1 and 2, Xj (Z) and X2 (Z) were used to define a residual r,... 

Density operator equations were converted into coupled integrodifferential equations suitable for numerical processing, and an extended Runge-Kutta algorithm has been implemented for solving the matrix equations in diadic form. A similar procedure can be followed for the original density matrix. 

Now pump-probe stimulated emission spectra can be determined on the basis of Eqs. (142)-(144). For this purpose, the coupled master equations are numerically solved and then using Eq. (128) pump-probe stimulated emission spectra are computed. For numerical simulation, the Runge-Kutta algorithm with a time step of 4 fsec is employed, and from 7 to 16... 

Coefficients au and b, are determined in order that the algorithm possesses some qualities such as stability, accuracy, etc. A classical explicit fourth-order Runge—Kutta algorithm is defined by the values... 

One-step algorithms have been developed and used by Prothero et al. [146, 166], Cdme et al. [156, 168], Pratt [177], Villadsen et al. [178] and Layokun and Slater [179]. Embedded semi-implicit Runge—Kutta algorithms have been discussed by Lapidus and co-workers [180]. 

Calculated effective diffusivities are tabulated in Table II. This problem was solved on the computer by the "shooting method" for a range of temperatures. Integrations were by the 4th order Runge-Kutta algorithm. The system equations are summarized below. 

The author gratefully acknowledges the assistance of R. Underwood and G. Johnson who guided the author in developing the Runge-Kutta algorithm. W. E. Sharp,... 

J. Vigo-Aguiar, J. Martin-Vaquero and H. Ramos, Exponential fitting BDF-Runge-Kutta algorithms. Computer Physics Communications, 2008, 178, 15-34. 

The temporal update through the single-stage ifth-order Runge-Kutta algorithms [33] is performed by... 

The Mathcad intrinsic function rkfixed provides the basic approach for solving numerically a system of first-order ODEs implementing the fourth-order Runge-Kutta algorithm. The call to this function has the form... 

The system of ordinary differential equations represented by Equation 45 can be integrated numerically with a fourth-order Runge-Kutta algorithm, as discussed by Carnahan and Wilkes [46]. 

In Fig. 5.48 the experimental measurements of the time dependence of fluorescence for carbol in oxygen free methanolic solution by irradiation at 254 nm are given for two wavelengths of observation. Using the iterative Runge-Kutta algorithm the parameters are fitted to the curve and the result is given as a theoretical curve as a line in Fig. 5.48. 

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