Transcript Intro to Runge-Kutta

Introduction to the Runge-Kutta
algorithm for approximating
derivatives
PHYS 361
Spring, 2011
review: Euler method
General form of first-oder Diff Eq: d x(t)  f (x,t)
dt
Example: decay equation
f (x,t)  
x

To implement a numerical solution, 
we must approximate the derivative.

The Euler method is the simplest approach: dx  x  x i1  x i
dt
t
t i1  t i
(forward difference)
Substituting this into our decay equation, we can solve for xi+1

xi1  xi  f (x,t)t
This is equivalent to linearly extrapolating x(t) from xi to xi+1
or... using the Taylor series expansion for x(t) and cutting out all terms
t2 or higher. This makes the Euler method “first order accurate”
involving
Taylor series
dx
1 d2x
2
x(t  t)  x(t) 
t 
t  ...
2  
dt
2 dt
Substitute f(x) for dx/dt (from our equation), keep up to 2nd order in dt

1  f f 
2
x(ti1 )  x(t i )  f it  f
 t 
2  x t i
This approach is fine if we know df/dx and df/dt

Runge-Kutta
make use of Mean value theorem
Same idea as Euler method, but aim for “exact” solution by using the
“mean value” slope, instead of the slope at ti.
xi1  xi  t  f (xm ,tm )
x
xm

ti
tm
Approximate the mean value slope
by evaluating f(x,t) at xm and tm
1
tm  t i  t
2
ti+1
Use Euler method to approximate xm: x i1/ 2  x i 
Decay example:
f (N,t)  
N
t
 f (x i ,ti )
2 
2nd order R-K has two steps
4th order R-K has four steps


Runge-Kutta... another approach
Approximate mean slope by averaging slopes at ti, ti+1, and ti+1/2
Decay example:
x
f (N,t)  
N
f1  f (xi ,ti )


xi
xm
xi+1

ti
tm
ti+1

Use weighted average of slopes:

t
x m1  x i 
f1
2

t
x m2  x i 
f2
 2
x
i1  x i  t  f 3

f 2  f (xm1,tm )
f 3  f (xm2,tm )
f 4  f ( x
i1,t i1)
t
x i1  x i   f
1  2 f2  2 f3  f4 
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