MATH 371 -- Numerical Analysis

Runge-Kutta Examples

November 26, 2001

The fourth-order Runge-Kutta method is a superior ODE method with

local truncation error . Each step of the method involves 4 evaluations

of the slope function as follows:

We illustrate this with the initial value problem

y' =

y(0) = 1

In this case there is an elementary solution (first order linear

equation):

> f:=(t,y) -> 3*t^2-y:

> dsolve({diff(y(t),t)=f(t,y(t)),y(0)=1});

> ansol:=t->3*t^2-6*t+6-5*exp(-t):

Applying Runge-Kutta to find approximate solution

using and 4 steps to get the solution at .

> T[0] := 0: RKW[0]:=1:

> h:=.25:

> for i to 4 do

> k[1]:=f(T[i-1],RKW[i-1])*h;

> k[2]:=f(T[i-1]+h/2,RKW[i-1]+k[1]/2)*h;

> k[3]:=f(T[i-1]+h/2,RKW[i-1]+k[2]/2)*h;

> k[4]:=f(T[i-1]+h,RKW[i-1]+k[3])*h;

> RKW[i]:=RKW[i-1]+(k[1]+2*k[2]+2*k[3]+k[4])/6;

> T[i]:=T[i-1]+h;

> end do:

> RKpts:=[[T[0],RKW[0]],[T[1],RKW[1]],[T[2],RKW[2]],[T[3],RKW[3]],[T[4],RKW[4]]]:

Compare with Euler's Method and the exact solution

> T[0]:=0: EulerW[0]:=1: h:=.25:

> for i to 4 do T[i]:=T[i-1] + h; EulerW[i]:=EulerW[i-1]+f(T[i-1],EulerW[i-1])*h; end do:

The exact and approximate solution values

t Exact solution Euler approx. Runge-Kutta approx.

0 1 1 1

.25 .793496084 .75 .7935180663

.50 .717346702 .609375 .7173944413

.75 .825667236 .64453125 .8257417646

1.00 1.160602794 .9052734375 1.160703425

Note the largest RK error is about .0001 (!)

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