Runge-Kutta formula

Runge-Kutta formulas of the sixth and eighth orders are also available, but less commonly used. 

This formula and the higher-order formulas are not self starting since they require f,, f etc. The common practice is to employ a Runge-Kutta formula of the same order to compute the first term(s) of y,. The general Adams open formula may be written as... 

J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. 

The Rimge-Kutta methods for numerical solution of the differential equation dy/dx = F(x, y) involve, in effect, the evaluation of the differential function at intermediate points between xi and Xj+i. The value of yi+ is obtained by appropriate summation of the intermediate terms in a single equation. The most widely used Runge-Kutta formula involves terms evaluated at X(, Xj + Ax/2 and X + Ax. The fourth-order Runge-Kutta equations for dy/dx = F(x, y) are... 

Fehlberg, E. Low-Order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems, NASA TR R-315 NASA Tech. Report 1969. 

Some recent Runge-Kutta formulae are based on quadrature methods, that is, the points at which the intermediate stage approximations are taken are the same points used in integration with either Gauss or Lobatto or Radau rules (Chapter 1). For example, the Runge-Kutta method derived from the Lobatto quadrature with three points (also called the Cavalieri-Simpson rule) is... 

Expand the Runge-Kutta formula for h and compare with the Taylor series in part (a) to show that one possible solution for w, W2, Cj, and is... 

Use the third order Runge-Kutta formula obtained in Problem 7.4 to derive the integration formulas for the coupled ordinary differential equations. 

Unfortunately, most explicit Runge-Kutta formulas have very poor convergence properties when used for half-explicit methods in the above way. In general they do not exceed order two. Exceptions are those sets of coefficients developed in [BH93]. We cite the coefficients of their third order method... 

EJNT86a] Enright W. H., Jackson K. R., N0rsett S. P., and Thomson P. G. (1986) Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants. Appl. Math, Comp, 27 313-335. 

The procedure for deriving the Runge-Kuttamethods can be divided into five steps which are demonstrated below in the derivation of the second-order Runge-Kutta formulas. 

With this set of pai ameters, the second-order Runge-Kutta formula is... 

Higher-order Runge-Kutta formulas are derived in an analogous manner. Several of these are listed in Table 5.2. The fourth-order Runge-Kutta, which has an error of O(h ), is probably the most widely used numerical integration method for ordinary differential equations. 

---