ODEs occur quite often in physics and astrophysics: Wave Equation in 1-D stellar structure equations hydrostatic equation in atmospheres orbits

Transcription

1 Solving ODEs

2 General Stuff ODEs occur quite often in physics and astrophysics: Wave Equation in 1-D stellar structure equations hydrostatic equation in atmospheres orbits need workhorse solvers to deal with them PDEs are more common

3 General Stuff these methods apply to all ODEs higher order ODEs are transformed to systems of first order ODEs, e.g., transforms to dy 1 dx dy 2 dx d 2 y dx 2 + q(x)dy dx = r(x) with y(x) = y 1 (x) and dy/dx = y 2 (x) = y 2 (x) (1) = r(x) q(x)y 2 (x) (2)

4 General Stuff need methods to solve a set of ODEs dy i dx = f i(x, y (1...i) ) equations of more than one independent variable PDEs require far more complex methods next chapter

5 Boundary Conditions need BCs to solve a ODE problem numerically analytical solution integration constants type of BCs determine numerical approach, e.g.: 1. initial value problem: all y i are given at one starting point x s 2. two-point boundary value problem: some y i are given at x s, the others at some point x f first: initial value problems

6 Euler method consider only ODE with i = 1 for simplicity! index y n value of y(x) at x = x n, i.e., the nth point going from x s to x f simplest approach: Euler chose stepsize h for x then write y n+1 = y n + hf (x n, y n ) uses information only at the beginning of an interval expansion in power series first order accuracy: error is O(h 2 )

7 Euler method

8 Euler method not recommended for practical use: Euler Predictor Corrector is better other methods have better accuracy for same step size instability problems (see below)

9 Euler Predictor-Corrector then slope of chord is y 1 = y 0 + d f d x x 0 h (3) y 0 = y 1 d f d x x 1 h (4) y 1 y 0 h = 1 2 {f (x 0, y 0 ) + f (x 1, y 1 )} but don t know y 1 use Euler step to estimate (predictor step) Y 1 = y 0 + f (x 0, y 0 ) h then use cord to advance the equation y 1 = y h{f (x 0, y 0 ) + f (x 1, Y 1 )}

10 Euler PC Method

11 Runge-Kutta method use Euler to take trial step to x n + h/2 use Euler estimate to approximate x and y at midpoint use these to compute full step midpoint method second order Runge-Kutta method

12 Midpoint Method

13 Runge-Kutta method formally: k 1 = hf (x n, y n ) (5) k 2 = hf (x n + h/2, y n + k 1 /2) (6) y n+1 = y n + k 2 + O(h 3 ) (7) second order accurate! needs 2 evals of f () for stepsize h can be extended by using several estimates at start/mid/endpoints accuracy can be increased

14 4th order RK method

15 Runge-Kutta method formally: k 1 = hf (x n, y n ) (8) k 2 = hf (x n + h/2, y n + k 1 /2) (9) k 3 = hf (x n + h/2, y n + k 2 /2) (10) k 4 = hf (x n + h, y n + k 3 ) (11) y n+1 = y n + k 1 /6 + k 3 /3 + k4/6 + O(h 5 ) (12) fourth order accurate better than 2nd order if h is at least 2 times larger for same quality

16 step size control very often f () changes significantly over integration interval to ensure homogeneous quality we need to adapt h needs to be really small in domains where f () has nasty features h can be larger for smooth, slowly varying functions

17 step size control general ODE solver needs to track accuracy to obtain best performance for given quality step-doubling: take each step twice as h and two h/2 steps total of 3 RK steps with 4 function evals each starting point shared 11 evals overhead is 11/ (we reach accuracy for step size h/2!)

18 step size control indicate truncation error difference between the two estimates use this information to adapt h automatically method is 4th order scales as h 5 ( ) h 0 = h 1 1 given a target truncation error 0, use this to estimate h 0 from current 1 and h 1 if 1 > 0 decrease h otherwise increase it

19 Fehlberg method RK of M > 4th order up to M + 2 evals 4th order most efficient? Fehlberg 5th order method with 6 evals these 6 points can also be used to build a 4th order formula combine them to compute truncation error with only 6 evals! embedded RK-Fehlberg method

20 Embedded RKF method

21 Embedded RKF method

22 Modified Midpoint Method go through a (large) step H with h/n sub-steps use an algorithm of the form z 0 = y(x) (13) z 1 = z 0 + hf (x, z 0 ) (14) z m+1 = z m 1 + 2hf (x + mh, z m ) (15) y(z + H) = 1 2 [z n + z n 1 + hf (x + H, z n )] (16) same as midpoint method but first and last points modified midpoint method

23 Modified Midpoint Method truncation error y n y(x + H) = i α i h 2i only even powers in h combining steps will deliver two orders better accuracy per doubling usually adaptive RKF s are better but modified midpoint is very useful for...

24 Richardson extrapolation

25 Bulirsch-Stoer method original Richardson power series limited convergence radius! Bulirsch-Stoer: use modified midpoint method use rational functions in h 2 can take rather large steps H

26 Bulirsch-Stoer method how to subdivide steps best? H/n j Bulirsch-Stoer: n j = 2n j 2 more efficient: n j = 2j (Deuflhard)

27 Bulirsch-Stoer method j max is not determined a priori extrapolation delivers error limits use them to limit steps variation: polynomial extrapolation

28 Bulirsch-Stoer method method works extremely well for well behaved ODEs smooth functions it does not work well for function evals through table lookup or interpolation discontinuous functions internal singular points steep changes of f through an interval

29 Predictor-Corrector Methods write the solution of the ODE as y n+1 = y n + xn+1 x n f (t, y) dt approximate f (t, y) by a polynomial through a number of previous points x n (?), x n 1, x n 2 multistep method result of has the form y n+1 = y n +h(β 0 f (x n+1, y n+1 )+β 1 f (x n, y n )+β 2 f (x n 1, y n 1 )+ )

30 Predictor-Corrector Methods β 0 = 0 explicit, otherwise implicit how to solve? Newton s method multistep solver for stiff problems

31 Predictor-Corrector Methods functional iteration predictor-corrector methods idea: use explicit method to get estimate y n+1 predictor step next: use estimate to correct y n+1 corrector step can be iterated if number of iterations is fixed at the beginning explicit method (why?)

32 Adams-Bashforth-Moulton method overall good stability (explicit!) 3rd order version: predictor step: Adams-Bashforth y n+1 = y n + h 12 corrector step: Adams-Moulton ( 23y n 16y n 1 + 5y n 2 ) + O(h 4 ) y n+1 = y n + h 12 ( 5y n+1 + 8y n y n 1 ) + O(h 4 )

33 predictor-corrector methods have starting/stopping problems (use RK) stepsize control hard use canned routines whenever possible historically very frequently used very good for very smooth functions that are expensive to evaluate

34 ODEINT Useful Workhorse from Num Rec Can be called with either RK4 (Runge-Kutta) OR with BSSTEP (Burlisch-Stoer)

35 My favorite method Hammings Modified Predictor Corrector Method Adaptive Stepsize control Written at IBM in the 1960s 4th order method Uses RK4 to start Will go exactly to b if initial stepsize is (b a)/2 n