Numerical Methods Contents - SAM

8 1/5 3/40 44/45 19372/6561 9017/3168 35/384
9 0 9/40 −56/15 −25360/2187 −355/33 0
10 0 0 32/9 64448/6561 46732/5247
500/1113
11 0 0 0 −212/729 49/176 125/192
12 0 0 0 0 −5103/18656
−2187/6784
13 0 0 0 0 0 11/84
14 0 0 0 0 0 0
15 ] ;
16 E = [71/57600; 0 ; −71/16695; 71/1920; −17253/339200; 22/525; −1/40];
17 % . (choice of stepsize and main loop omitted)
18 % ADVANCING ONE STEP.
19 hA = h ∗ A ;
20 hB = h ∗ B ;
21 f ( : , 2 ) = feval ( odeFcn , t +hA ( 1 ) , y+ f ∗hB ( : , 1 ) , odeArgs { : } ) ;
22 f ( : , 3 ) = feval ( odeFcn , t +hA ( 2 ) , y+ f ∗hB ( : , 2 ) , odeArgs { : } ) ;
23 f ( : , 4 ) = feval ( odeFcn , t +hA ( 3 ) , y+ f ∗hB ( : , 3 ) , odeArgs { : } ) ;
24 f ( : , 5 ) = feval ( odeFcn , t +hA ( 4 ) , y+ f ∗hB ( : , 4 ) , odeArgs { : } ) ;
25 f ( : , 6 ) = feval ( odeFcn , t +hA ( 5 ) , y+ f ∗hB ( : , 5 ) , odeArgs { : } ) ;
Ôº ½½º
29 h = tnew − t ; % Purify h.
26
27 tnew = t + hA ( 6 ) ;
28 i f done , tnew = t f i n a l ; end % Hit end point exactly.
Special case of oscillating Zhabotinski-Belousov reaction [19]:
BrO − 3 + Br− ↦→ HBrO 2
HBrO 2 + Br − ↦→ Org
BrO − 3 + HBrO 2 ↦→ 2 HBrO 2 + Ce(IV)
(11.5.1)
2 HBrO 2 ↦→ Org
Ce(IV) ↦→ Br −
y 1 := c(BrO − 3 ): ẏ 1 = −k 1 y 1 y 2 − k 3 y 1 y 3 ,
y 2 := c(Br − ): ẏ 2 = −k 1 y 1 y 2 − k 2 y 2 y 3 + k 5 y 5 ,
y 3 := c(HBrO 2 ): ẏ 3 = k 1 y 1 y 2 − k 2 y 2 y 3 + k 3 y 1 y 3 − 2k 4 y3 2 ,
y 4 := c(Org): ẏ 4 = k 2 y 2 y 3 + k 4 y3 2 ,
y 5 := c(Ce(IV)): ẏ 5 = k 3 y 1 y 3 − k 5 y 5 ,
with (non-dimensionalized) reaction constants:
k 1 = 1.34 , k 2 = 1.6 · 10 9 , k 3 = 8.0 · 10 3 , k 4 = 4.0 · 10 7 , k 5 = 1.0 .
periodic chemical reaction ➽ Video 1, Video 2
(11.5.2)
MATLAB simulation with initial state y 1 (0) = 0.06, y 2 (0) = 0.33 · 10 −6 , y 3 (0) = 0.501 · 10 −10 ,
y 4 (0) = 0.03, y 5 (0) = 0.24 · 10 −7 :
Ôº½ ½½º
30 ynew = y + f ∗hB ( : , 6 ) ;
31 % . (stepsize control, see Sect. 11.5 dropped
10 −3 Concentration of Br −
10 −4
10 −6
△
10 −5
10 −7
Example 11.4.7 (Numerical integration of logistic ODE in MATLAB).
10 −6
MATLAB-CODE: usage of ode45
fn = @(t,y) 5*y*(1-y);
[t,y] = ode45(fn,[0 1.5],y0);
plot(t,y,’r-’);
MATLAB-integrator: ode45():
Handle passing r.h.s.
initial and final time
initial state y 0
✸
c(t)
10 −7
10 −8
10 −9
10 −10
0 20 40 60 80 100 120 140 160 180 200
Fig. 144
t
c(t)
10 −5 t
10 −8
10 −9
10 −10
10 −11
Concentration of HBrO 2
Fig. 145
0 20 40 60 80 100 120 140 160 180 200
We observe a strongly non-uniform behavior of the solution in time.
11.5 Stepsize control
Example 11.5.1 (Oregonator reaction).
Ôº¼ ½½º
This is very common with evolutions arising from practical models (circuit models, chemical reaction
models, mechanical systems)
Ôº¾ ½½º