Numerical Methods Contents - SAM

Example 11.5.6 (Gain through adaptivity). → Ex. 11.5.4
Simple adaptive timestepping from previous experiment Ex. 11.5.4.
New: initial state y(0) = 0!
Now we study the dependence of the maximal point error on the computational effort, which is proportional
to the number of timesteps.
Code 11.5.7: MATLAB function for Ex. 11.5.6
1 function adaptgain ( T , a , r e l t o l , a b s t o l )
2 % Experimental study of gasin through simple adaptive timestepping
3 % strategy of Code 11.5.2 based on explicit Euler
4 % (11.2.1) and explicit trapezoidal
5 % rule (11.4.3)
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9 i f ( nargin < 3) , r e l t o l = 1E−1; end
6
7 % Default arguments
8 i f ( nargin < 4) , a b s t o l = 1E−3; end
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33 end
34
35 % Run adaptive timestepping with various tolerances, which is the only way
36 % to make it use a different total number of timesteps.
37 % Plot the solution sequences for different values of the relative tolerance.
38 figure ( ’name ’ , ’ adaptive timestepping ’ ) ;
39 axis ( [ 0 2 0 0 . 0 5 ] ) ; hold on ; c o l = colormap ;
40 errad = [ ] ; l = 1 ;
41 for r t o l = r e l t o l ∗2.^(2: −1: −4)
42 % Crude choice of initial timestep and h min
43 h0 = T / 1 0 ; hmin = h0 /10000;
44 % Main adaptive timestepping loop, see Code 11.5.2
45 [ t , y , r e j , ee ] =
odeintadapt_ext ( Psilow , Psihigh , T , y0 , h0 , r t o l ,0.01∗ r t o l , hmin ) ;
46 errad = [ errad ; length ( t ) −1, max( abs ( s o l ( t ) − y ) ) , r t o l ,
length ( r e j ) ] ;
47 f p r i n t f ( ’ r t o l = %d : %d timesteps , %d r e j e c t e d timesteps \ n ’ ,
r t o l , length ( t ) −1,length ( r e j ) ) ;
48 plot ( t , y , ’ . ’ , ’ c o l o r ’ , c o l (10∗( l −1) + 1 , : ) ) ;
49 leg { l } = s p r i n t f ( ’ r t o l = %f ’ , r t o l ) ; l = l +1;
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50 end
51 xlabel ( ’ { \ b f t } ’ , ’ f o n t s i z e ’ ,14) ;
52 ylabel ( ’ { \ b f y } ’ , ’ f o n t s i z e ’ ,14) ;
10 i f ( nargin < 2) , a = 40; end
11 i f ( nargin < 1) , T = 2 ; end
12
13 % autonomous ODE ẏ = cos(ay) and its general solution
14 f = @( y ) ( cos ( a∗y ) . ^ 2 ) ; s o l = @( t ) ( atan ( a∗( t ) ) / a ) ;
15 % Initial state y 0
16 y0 = s o l ( 0 ) ;
17
18 % Discrete evolution operators, see Def. 11.2.1
19 Psilow = @( h , y ) ( y + h∗ f ( y ) ) ; % Explicit Euler (11.2.1)
20 % Explicit trapzoidal rule (11.4.3)
21 Psihigh = @( h , y ) ( y + 0.5∗h∗( f ( y ) + f ( y+h∗ f ( y ) ) ) ) ;
22
23 % Lop over uniform timesteps of varying length and integrate ODE by explicit trapezoidal
24 % rule (11.4.3)
25 e r r u f = [ ] ;
26 for N=10:10:200
27 h = T /N; t = 0 ; y = y0 ; e r r = 0 ;
28 for k =1:N
29 y = Psihigh ( h , y ) ; t = t +h ;
30 e r r = max( err , abs ( s o l ( t ) −
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y ) ) ;
31 end
32 e r r u f = [ e r r u f ;N, e r r ] ;
53 legend ( leg , ’ l o c a t i o n ’ , ’ southeast ’ ) ;
54 t i t l e ( s p r i n t f ( ’ Solving d_t y = a cos ( y ) ^2 w i th a = %f by simple
adaptive timestepping ’ , a ) ) ;
55 p r i n t −depsc2 ’ . . / PICTURES/ adaptgainsol . eps ’ ;
56
57 % Plotting the errors vs. the number of timesteps
58 figure ( ’name ’ , ’ gain by a d a p t i v i t y ’ ) ;
59 loglog ( e r r u f ( : , 1 ) , e r r u f ( : , 2 ) , ’ r + ’ , errad ( : , 1 ) , errad ( : , 2 ) , ’m∗ ’ ) ;
60 xlabel ( ’ { \ b f no . N o f timesteps } ’ , ’ f o n t s i z e ’ ,14) ;
61 ylabel ( ’ { \ b f max_k | y ( t_k )−y_k | } ’ , ’ f o n t s i z e ’ ,14) ;
62 t i t l e ( s p r i n t f ( ’ E r r o r vs . no . o f timesteps f o r d_t y = a cos ( y ) ^2 w i th a
= %f ’ , a ) ) ;
63 legend ( ’ uniform timestep ’ , ’ adaptive timestep ’ , ’ l o c a t i o n ’ , ’ n o r theast ’ ) ;
64 p r i n t −depsc2 ’ . . / PICTURES/ adaptgain . eps ’ ;
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