RK4

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.

Syntax

Result = RK4( Y, Dydx, X, H, Derivs [, /DOUBLE] )

Return Value

Returns the integrations of the ordinary differential equations.

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:

dy₀ / dx = -0.5y₀_,dy₁ / dx = 4.0 - 0.3y₁ - 0.1y₀

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.

Examples

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

; Define the step size: 
H = 0.5 
 
; Define an initial X value: 
X = 0.0 
 
; Define initial Y values: 
Y = [4.0, 6.0] 
 
; Calculate the initial derivative values: 
dydx = DIFFERENTIAL(X,Y) 
 
; Integrate over the interval (0, 0.5): 
result = RK4(Y, dydx, X, H, 'differential') 
 
; Print the result: 
PRINT, result

IDL prints:

 3.11523  6.85767

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

Version History

Introduced: 4.0

RK4