RK4

The RK4 function uses the fourth-order Runge-Kutta method to advance a solution to a system of ordinary differential equations one time-step H , given values for the variables Y and their derivatives Dydx known at X .

RK4 is based on the routine ``` rk4``` described in section 16.1 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.

Calling Sequence

Result = RK4( Y, Dydx, X, H, Derivs )

Arguments

Y

A vector of values for Y at X

Dydx

A vector of derivatives for Y at X .

X

A scalar value for the initial condition.

H

A scalar value giving interval length or step size.

Derivs

A scalar string specifying the name of a user-supplied IDL function that calculates the values of the derivatives Dydx at X . This function must accept two arguments: A scalar floating value X , and one n -element vector Y . It must return an n -element vector result.

For example, suppose the values of the derivatives are defined by the following relations:

dy 0 / dx = -0.5 y 0,         dy 1 / dx = 4.0 - 0.3 y 1 - 0.1 y 0

We can write a function DIFFERENTIAL to express these relationships in the IDL language:

FUNCTION differential, X, Y

RETURN, [-0.5 * Y[0], 4.0 - 0.3 * Y[1] - 0.1 * Y[0]]

END

Keywords

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

Example

To integrate the example system of differential equations for one time step, H:

H = 0.5 ; Define the step size.

X = 0.0 ; Define an initial X value.

Y = [4.0, 6.0] ; Define initial Y values.

dydx = DIFFERENTIAL(X,Y) ; Calculate the initial derivative values.

result = RK4(Y, dydx, X, H, 'differential')
; Integrate over the interval (0, 0.5).

PRINT, result ; Print the result.

IDL prints:

3.11523  6.85767

This is the exact solution vector to five-decimal precision.