Runge-Kutta Method - Mathematica Implementation Part 1

Numerical Methods for Solving Differential Equations

The Runge-Kutta Method

MathematicaÌý±õ³¾±è±ô±ð³¾±ð²Ô³Ù²¹³Ù¾±´Ç²Ô

(continued fromÌýlast page...)

Recall from the first numerical methods lab that we had managed to create a program for finding numerical solutions of a first order differential equation usingÌýEuler's method. The program we created was as follows:

euler[f_,{x_,x0_,xn_},{y_,y0_},steps_]:=

Block[{ xold=x0,
ÌýÌýÌýÌý²â´Ç±ô»å=²â0,
ÌýÌýÌýÌý²õ´Ç±ô±ô¾±²õ³Ù=µ÷µ÷³æ0,²â0°¨°¨,
ÌýÌýÌýÌý³æ,²â,³ó
ÌýÌý°¨,

ÌýÌý³ó=±·°Ú(³æ²Ô-³æ0)/²õ³Ù±ð±è²õ±Õ;

ÌýÌýDo[ xnew=xold+h;
ÌýÌýÌýÌýynew=yold+h*(f/.{x->xold,y->yold});
ÌýÌýÌýÌýsollist=Append[sollist,{xnew,ynew}];
ÌýÌýÌýÌýxold=xnew;
ÌýÌýÌýÌýyold=ynew,
ÌýÌýÌýÌýsteps
ÌýÌý±Õ;

ÌýÌý¸é±ð³Ù³Ü°ù²Ô°Ú²õ´Ç±ô±ô¾±²õ³Ù±Õ
]

Ìý ÌýÌý Ìý Go ahead and switch to a newÌýMathematicaÌýnotebook by clicking on the button at left. Remember that it will take a while to start up! Once it has launched, enter the old Euler program above, (by usingÌýCopyÌýandÌýPasteÌýif you wish,) and then come back here again where we'll discuss modifying it.

Notice that I highlighted a couple of locations in the code inÌýred. Let'sÌýmove onÌýand discuss why...

Compass If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.