Backward Runge-Kutta Method (4th Order)

[wolfandraven]

Homework Statement

Hi,

As part of a computing project I'm doing, I need to use a shooting method, to solve a set of differential equations. I have some trial boundary conditions at either end of the equations, and am going to be integrating from each end and varying some parameters until the solutions meet in the middle.

I'd like to use the 4th order Runge-Kutta method to do this integration, but I only know the forwards Runge-Kutta method, and not the backwards one, so I'd just like to ask a few questions:

Firstly, is there a backwards version?

Second, what is it? Please could you state it, give me a link to it, or try and explain how I'd find it (as I'm having difficulty working it out).

Thanks!

Homework Equations

The 4th Order Forwards Runge-Kutta method is:

for dy/dx = F(X,Y)

Yn+1 = Yn + 1/6H(K1 + 2K2 + 2K3 + K4)

where

K1 = F(X,Y)*H

K2 = F(X+H/2,Y+H/2*K1)

K3 = F(X+H/2, Y+H/2*K2)

K4 = F(X+H, Y+H*K3)

The Attempt at a Solution

 

[lippyka]

Yes, there is a backwards version of the 4th order Runge-Kutta method. The equation for the backwards Runge-Kutta method is: Yn-1 = Yn - 1/6H(K1 + 2K2 + 2K3 + K4)whereK1 = F(X,Y)*HK2 = F(X-H/2,Y-H/2*K1)K3 = F(X-H/2, Y-H/2*K2)K4 = F(X-H, Y-H*K3)