3469 - Practical Numerical Simulations
Mike Peardon — mjp@maths.tcd.ie
http://www.maths.tcd.ie/~mjp/3469
School of Mathematics
Trinity College Dublin
Michaelmas Term 2015
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
1/8
Numerical solutions to ODEs
2. Boundary-value problems
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
2/8
Boundary-value problems
So far, have considered solving ODEs with boundary
conditions of the form
(0)
(1)
(2)
x(t0 ) = x0 , ẋ(t0 ) = x0 , ẍ(t0 ) = x0 , . . .
Techniques start with these values, then propagate the
solution to t0 + h, t0 + 2h, t0 + 3h, . . . .
Consider now case where the N boundary data are specified
at two distinct places, t0 , t1 .
A general boundary value problem
Solve an Nth-order ODE (with x (0) = x, x (1) = ẋ, . . . ) with boundary data of the form
ga (x(t0 )) = 0 for a = 1 . . . n
hb (x(t1 )) = 0 for b = n . . . N
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
3/8
Shooting method
Do not now have the data to start our iterative scheme.
Example: second-order ODE with x given at two end-points
x is known at two distinct points, t0 and t1 , so
x(t0 ) = x0 , x(t1 ) = x1
Initial-value solver needs data on ẋ(t0 ) to start, so guess a
value: ẋ(t0 ) = c
Now we have initial value data (although probably solving the
wrong problem!), so can use Euler, Runge-Kutta etc.
Solve to find x̄(t1 ; c), the approximate solution at t1 given c.
Determine the value of c for which the other boundary
condition x(t1 ) = x1 holds.
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
4/8
Finding consistent initial data
For general case, need N − n “fake” initial values for c
Will need to solve N − n equations for these unknowns.
In general, don’t known how many solutions there are (and
there could be none).
Finding the N − n consistent data can be challenging: use
Newton-Rapheson or bisection for a single missing value, c.
Newton-Rapheson needs information about dx/dc, which
might be hard to come by.
Bisection can miss solutions, needs help with good guesses
for locations of solutions, etc.
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
5/8
Example from Stoer and Bulirsch
Boundary-value 2nd-order ODE
3 2
x
2
with x(0) = 4, x(1) = 1
ẍ =
Solve instead the ODE with initial data:
x(0) = 4, ẋ(0) = c
and examine b(c) = x(1; c) − 1 for zeroes.
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
6/8
Example from Stoer and Bulirsch
60
b(c)
40
20
0
-100
-80
-60
-40
Guessed initial data, c
-20
0
Find two consistent solutions, c ≈ −35.9, −8.0
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
7/8
Example from Stoer and Bulirsch
The two solutions are then:
5
x(0)=4
x(1)=1
x(t)
0
-5
-10
0
0.2
0.4
0.6
0.8
1
t
Mike Peardon — mjp@maths.tcd.ie http://www.maths.tcd.ie/~mjp/3469
3469 (TCD)
Michaelmas Term 2015
8/8