Chapter 21Chapter 21

ODEs: Adaptive Methods ODEs: Adaptive Methods and Stiff Systemsand Stiff Systems

Adaptive Runge Kutta MethodAdaptive Runge Kutta Method

Use small step size in high gradient region

(abrupt change)

automatic step size adjustment

Adaptive Runge Kutta MethodAdaptive Runge Kutta Method

First approach: Step halvingEstimate local truncation error using two different

step sizes Solve each step twice, once as a full step and then

as two half steps

Second approach: Embedded RK methods (also called RK-Fehlberg methods)Estimate local truncation error between two

predictions using two different-order RK methods

Step Halving MethodStep Halving Method

Step Halving (Adaptive RK) method

Compute the solutions at each step twice using 4th-order classic RK method

Once as a full step h and independently as two half steps (h/2)

y1 – one full step; y2 – two half steps

15yy

yy

22

12

Error estimateError estimate

Correction – 5Correction – 5thth-order-order

Adaptive 4Adaptive 4thth-order RK Method-order RK Method

One full-step with h

Two half-steps with h/2

Therefore,

)()( 651 hOhyhxy

)()( 65

2 hO2

h2yhxy

15y

15

yyy

2

h2yy

yyh15

16 h

16

15

32

h2hyy

212

2

5

2

1255

55

12

)(

Two half-steps

Embedded Runge-Kutta MethodEmbedded Runge-Kutta Method

MATLAB function: ODE23MATLAB function: ODE23 BS23 algorithm (Bogacki and Shampine, 1989; Shampine, 1994)

Use 3rd-order & 4th-order RK methods simultaneously to solve the ODE and estimate the error for step-size adjustment

Error estimate (Note: k1 is the same as k4 from previous step)

),(

),(

),(

)(

hk4

3yh

4

3xfk

hk2

1yh

2

1tfk

ytfk

;hk4k3k29

1yy

2ii3

1ii2

ii1

321i1i

),()( 1i1i443211i ytfk ;hk9k8k6k572

1E

Embedded RK Method: ODE23Embedded RK Method: ODE23

Uses only three function evaluations (k1, k2, k3)

After each step, the error is checked to determine whether it is within desired tolerance. If it is, yi+1 is accepted and k4 becomes k1 for the next time step

If the error is too large, the step is repeated with a reduced step sizes until the estimate error is acceptable

RelTol: relative tolerance (default = 103)

AbsTol: relative tolerance (default = 106)

),max( AbsTolyRelTolE

Adaptive RK Method – ode23Adaptive RK Method – ode23

Example 21.2: Use ode23 to solve the following ODE from t = 0 to 4:

500y e10y60dt

dy 22 075022t .)(;. ]).(/[)(

function yp = ex21_2(t, y)

% Example 21.2 in the text book

yp = 10*exp(-(t-2)*(t-2)/(2*0.075^2)) - 0.6*y;

Example 20.2: ode23Example 20.2: ode23>> options = odeset('RelTol',1.e-4);

>> ode23('ex21_2', [0 4], 0.5, options);

>> options = odeset('RelTol',1.e-3);

>> ode23('ex21_2', [0 4], 0.5, options);

(a) RelTol = 103 (b) RelTol = 104

Other MATLAB FunctionsOther MATLAB Functions

MATLAB Function: ode45 Dormand and Prince (1990)

Solve fourth- and fifth-order RK formulas simultaneously

Make error estimates for step-size adjustment

MATLAB Function: ode113

Adams-Bashforth-Moulton solver (order 1-12)

Predictor-corrector method

Runge-Kutta Fehlberg MethodRunge-Kutta Fehlberg Method

Fourth-orderFourth-order

Fifth-order Fifth-order

1

2 1

3 1 2

4 1 2 3

5 1 2 3 4

6 1 2 3

( , )

1 1( , )

5 53 3 9

( , )10 40 40

3 3 9 6( , )

5 10 10 511 5 70 35

( , )54 2 27 27

7 1631 175 575 44275( ,

8 55296 512 13824 11

i i

i i

i i

i i

i i

i i

k f x y

k f x h y k h

k f x h y k h k h

k f x h y k h k h k h

k f x h y k h k h k h k h

k f x h y k h k h k h

4 5

253)

0592 4096k h k h

hk1771

512k

594

125k

621

250k

378

37yy 6431i1i )(

hk4

1k

14336

277k

55296

13525k

48384

18575k

27648

2825yy 65431i1i )(

These coefficients were developed by Cash and Karp (1990). Also called Cash-Karp RK Method

Runge-Kutta Fehlberg MethodRunge-Kutta Fehlberg Method

Identical coefficients (k1, k2, k3, k4, k5, k6) for both the

fourth and fifth order Runge-Kutta-Fehlberg methodsSave CPU time

Error estimate – use two RK methods of different order to estimate the local truncation error

hk1771

512

4

1hk 0

14336

277hk

594

125

55296

13525

hk 621

250

48384

18575hk 00hk

378

37

27648

2825Error

654

321

Runge-Kutta Fehlberg MethodRunge-Kutta Fehlberg Method

First, calculate yi+1 using 4th-order Runge-Kutta

Felberg method (y1)4th

Then, calculate yi+1 using 5th-order Runge-Kutta

Felberg method (y2)5th

Calculate Error Ea = (y2)5th - (y1)4th

Adjust step size according to error estimate

Step Size ControlStep Size Control

Use step-halving or Runge-Kutta Fehlberg to estimate the local truncation error

Adjust the step size according to error estimate Increase the step size if the error is too small and

decrease it if the error is too largeFor fourth-order schemes

present

present

0.2 if ;

0.25 if

Desired error value

newnewnew present

newpresent

new

h h

Step Size AdjustmentStep Size Adjustment

For step size increases (RK4, n = 4)

For step size decreases, h is implicit in new

h

h

present

new

1n

present

new

20

present

newpresent

1n

1

present

newpresentnew hhh

.

For nth-order RK Method

250

present

newpresent

n

1

present

newpresentnew hhh

.

Reduce hnew also

reduce new

ExampleExample21.221.2

Forcing function

solution

0.5y(0) e10y60dt

dy 22 075022t ;. ]).(/[)(

small step size around t = 2

Runge-Kutta-Fehlberg method

with adaptive step size control

t

t

Predator-Prey EquationPredator-Prey Equation

A simple predator-prey relationship is described by the Lotka-Volterra model, which we write in terms of a fox population f(t), with birth rate bf and death rate df and a geese population g(t) with birth rate bg and death rate dg

)(

)(

)()(

)()(

1gg22

f2f11

gg

ff

ydbydt

dy

dybydt

dy

or

tfdbtgdt

dg

dtgbtfdt

df

Predator-Prey EquationPredator-Prey Equation

Example: Given bf = 0.3, df = 0.8, bg = 1.2, dg = 0.6, find the populations of predators and preys as a function of time (t = 0 to 20) using ode45.

20y yy60y21dt

dy

10y y80yy30dt

dy

12122

21211

)(;..

)(;.. Predator

Prey

function yp = predprey(t,y)% Fox (Predator) population y1(t), birth rate bf, death rate df% Geese (prey) poupulation y2(t), birth rate bg, death rate dgbf = 0.3; df = 0.8; bg = 1.2; dg = 0.6;yp = [y(1)*(bf*y(2)-df); y(2)*(bg-dg*y(1))];

>> tspan=[0 20]; y0 = [1, 2];>> [t,y] = ode45('predprey',tspan,y0);>> out = [t y]out = 0 1.00000000000000 2.00000000000000 0.08372954771699 0.98466335925883 2.10387606351447 0.16745909543397 0.97217906418030 2.21469691968449 0.25118864315096 0.96261463455261 2.33264806768044 0.33491819086794 0.95606000043195 2.45787517860514 0.62870201024789 0.95940830414765 2.95257534365817 0.92248582962784 1.00915715534212 3.53554512325941 1.21626964900779 1.11944001342999 4.17743733245175 1.51005346838774 1.31420023093313 4.80288705430266 1.70029730481714 1.49877617603179 5.14418160032910 1.89054114124654 1.73932623590652 5.37426899115905 2.08078497767594 2.03724630610217 5.44284022577897 2.27102881410534 2.38123550265509 5.31619117469119 2.46127265053474 2.74867929767025 4.98122010458805 2.65151648696414 3.09373931102751 4.48531246964154 2.84176032339354 3.37201992118282 3.89965751560944 3.03200415982294 3.55414367377695 3.29772307544844 3.21873982743099 3.62508046596480 2.75627545562992 3.40547549503903 3.59592495497541 2.29786856084044 3.59221116264708 3.48499741488926 1.93369109247072 3.77894683025512 3.31638942318525 1.65488750920928 3.94714350340851 3.13529220765148 1.46223282653043 4.11534017656189 2.93901369774356 1.31685686503153 4.28353684971528 2.73779832900552 1.20974621108398 4.45173352286867 2.53877488937042 1.13408335792525 4.65875081398499 2.30372716989349 1.07647855767312 4.86576810510130 2.08503830477357 1.05104469525303 5.07278539621762 1.88590370110986 1.05298367705029 ... ... ... 19.13258672508553 1.39621015109817 1.21506308763510 19.36078504002272 1.26894875046392 1.33168491347240 19.58898335495991 1.16392668931057 1.48284270918163 19.69173751621993 1.12358612912238 1.56325743063018 19.79449167747995 1.08747228241032 1.65192298666833 19.89724583873998 1.05554028549720 1.74928308667197 20.00000000000000 1.02776980338246 1.85579178601841

Adaptive RK method : ode45Adaptive RK method : ode45

Predator-Prey ModelPredator-Prey Model

Time history of predator (fox) and prey (geese) populations

Predator-Prey ModelPredator-Prey Model

State-space plot

Multistep MethodsMultistep Methods

Runge-Kutta methods -- one-step method

-- use intermediate values between ti and ti+1

-- several evaluations of slope per step

Multistep methods -- use values at ti , ti-1 , ti-2 etc

-- only one evaluation of derivative per step

One-Step and Multistep MethodsOne-Step and Multistep Methods

One-step Multistep

- explicit (b0 = 0) & implicit methods. - # of the previous steps.

• Non-self start Huen method*

•Adams-Bashforth Methods (explicit methods b0 = 0)

•Admas-Moulton Methods (implicit methods)

•Predictor-Corrector Methods

] ...),( ),(

),([..

1121

1101211

iiii

iiiii

yxfbyxfb

yxfbhyayay

Multi-step methods use several previous points (yi-1 , yi-2 ,…) in addition to the present point yi

Start with second-order method

Look at Heun’s method (Self-Starting)

Euler predictor - O(h2)

Trapezoid corrector - O(h3)

hytfyy iii0

1i ,

h

2

ytfytfyy

01i1iii

i1i

,,

Heun’s Method Heun’s Method (Review)(Review)

To improve predictor, use 3rd-order Euler method

along with the same old corrector

iterated until converged

)))(,( 3mii

m1i

01i O(h ;h2ytfyy

)(),(),(

m,1,2,j ;h2

ytfytfyy

1j1i1i

miim

ij

1i

sj1i

1j1i

j1i

a %100y

yy

Non-Self-Starting Heun’s MethodNon-Self-Starting Heun’s Method

mi

m1i yy ,Note: are the final results of the corrector iterations

at the previous time step

Non-Self Starting Heun’s MethodNon-Self Starting Heun’s Method

(a) 3(a) 3rdrd-order predictor (b) 3-order predictor (b) 3rdrd-order corrector-order corrector

Non-self starting, need yi-1 and yi (two initial values)

Heun’s MethodHeun’s Method

(a) 2(a) 2ndnd-order predictor (b) 3-order predictor (b) 3rdrd-order corrector-order corrector

Self-starting, need yi only

Truncation ErrorsTruncation Errors

Non-self-starting Heun methodNon-self-starting Heun method

Predictor truncation errorPredictor truncation error

Corrector truncation errorCorrector truncation error )(

)(

C3m

1i

P30

1i

yh12

1y Value True

yh3

1y Value True

)()(

,)()(

yh12

5yh

3

1

12

1yy

then yy If

33m1i

01i

CP

ModifiersModifiers Corrector Modifier

Predictor Modifier (not used in textbook)

5

yyyy

5

yyyh

12

1E

01i

m1im

1im

1i

01i

m1i3

C

)(

)(

)()(

0i

mi

01i

01i

0i

mi

3P

yy5

4yy

yy5

4yh

3

1E

Error in textbook

So the sequence is

Predict

Adjust prediction*

Correct

Converged?

If not, correct again

Non-Self Starting Heun’s MethodNon-Self Starting Heun’s Method

Hand CalculationsHand Calculations Non-self-starting Heun’s method

Need two initial conditions

Example: 21.3 (p523- 524)

0.80.5 4 ( , )

(0) 2, y(-1.0) 0.3929953

Find (1.0) to (4.0) using h 1.0

tdyy e f t y

dty

y y

Exact solution: y = 3.076923e 0.8t – 1.076923 e –0.5 t

Hand Calculations – First StepHand Calculations – First Step

First Step (i = 0) : need two initial conditions

Predictor:

h2

ytfytfyy

h2ytfyy1j

1i1imiim

ij

1i

mii

m1i

01i

),(),(

))(,(

Predictor:

Corrector:

1 0 1

1 0

1.0, 0, 1.0;

0.3929953, 2.0;m m

t t t

y y

0 0.8*00 0 0

01 1 0 0

( , ) 0.5 0.5(2) 4 3

( , )(2 ) 0.3929953 (3)(2)(1) 5.06070

tm m

m m

f t y y e e

y y f t y h

h2ytfyy m00

m1

01 ))(,(

Hand Calculations – First StepHand Calculations – First Step Corrector:

1

1

0.80 0 0.8*11 1 1

11

0.81 1 0.81 1 1

21

( , ) 0.5 4 0.5(5.607005) 4 6.098661

3 6.0986612 (1) 6.549331

2

( , ) 0.5 4 0.5(6.549331) 4 5.627498

3 5.6274982 (1) 6.313749

2

t

t

f t y y e e

y

f t y y e e

y

h2

ytfytfyy

1j11

m00m

0j

1

),(),(

1Exact 6.194631 , 1.9%, 3.7%et ey

10.82 2 0.81 1 1

31

1

( , ) 0.5 4 0.5(6.313749) 4 5.745289

3 5.7452892 (1) 6.372645, 0.9%,

2

After many iterations, 6.36087

t

e

m

f t y y e e

y

y

Hand Calculations – Second StepHand Calculations – Second Step

Second Step (i = 1) :

Predictor:

0 1 2

0 1

0.0, 1.0, 2.0

2, 6.36087m m

t t t

y y

10.8 0.81 1 1

02 0 1 1

( , ) 0.5 4 0.5(6.36087) 4 5.72173

( , )(2 ) 2 (5.72173)(2)(1) 13.44346

tm m

m m

f x y y e e

y y f x y h

h2yxfyy m11

m0

02 ))(,(

Hand Calculations – First StepHand Calculations – First Step Corrector:

2

1

0.80 0 0.8*22 2 2

12

0.82 1 1.61 1 2

22

( , ) 0.5 4 0.5(13.44346) 4 13.09040

5.72173 13.090406.36087 (1) 15.76693

2

( , ) 0.5 4 0.5(15.76693) 4 11.92866

5.72173 11.928666.36087 (1) 15.18607

2

t

t

f t y y e e

y

f t y y e e

y

11 1 2 2

2 1

( , ) ( , )

2

m jj m f t y f t y

y y h

1Exact 14.84392 , ey

StiffnessStiffness A stiff system involves rapidly changing components (usually

die away quickly) together with slowly ones Long-time solution is dominated by slow varying components

tt1000exact

t

e0022e99803y

00y

e20003000y1000dt

dy

..

)(

Numerical StabilityNumerical Stability

Amplification or decay of numerical errors A numerical method is stable if error incurred at

one stage of the process do not tend to magnify at later stages

Ill-conditioned differential equation

-- numerical errors will be magnified regardless of numerical method Stiff differential equation

-- require extremely small step size to achieve accurate results

StabilityStability

Example problem eyy

y0y

aydt

dyat

0

0

)(

)(

/

)(,

0E

aEdtdE y(x)(x)yE(x) Error let

eyy then yy(0) if at00

stablelllyexponentia decay error :0a

stable neutrally :0a

unstable llyexponentia grow error :0a

eE at

Euler Explicit MethodEuler Explicit Method

Stability criterion

Region of absolute stability

iiii

i1i

0

yah1hayyhdt

yd yy

y0y ayyxfdt

yd

)()(

)(;),(

1ah1 ro 1y

y

i

1i

2ah0 1ah11

Amplification factor

Euler Implicit MethodEuler Implicit Method

Unconditionally stable !

ah1

yy

aydt

yd ;h

dt

yd yy

y0y ayyxfdt

yd

i1i

1i1i1i

i1i

0

)(;),(

h all orf 1ah1

1

y

y

i

1i

Forward,Forward,Backward,Backward,or Centeredor CenteredScheme?Scheme?

StabilityStability Explicit Euler method

Second-order Adams-Moulton

Implicit methods are in general more stable than explicit methods.

2ah0

ah0 Unconditionally stable

If a = 1000, then h 0.002 to ensure stability

Example Example 21.521.5

Euler Explicit

Euler Implicit

tt1000exact

t

e0022e99803y

00y ;e20003000y1000dt

dy

..

)(

MATLAB Functions for Stiff SystemsMATLAB Functions for Stiff Systems

ode15s – based on Gear backward differentiation

for stiff problems of low to medium accuracyode23s – based on modified Rosenbrock formulaode23t – trapezoidal rule with a free interpolant

for moderately stiff problems without

numerical dampingode23tb – implicit Runge-Kutta formula

for more explanation please see page 529

Stiff ODE – van del Pol equationStiff ODE – van del Pol equation

Van der Pol equation for electronic circuit

Convert to two first-order ODEs

1dt

0dy

10y sCI 0y

dt

dyy1

dt

yd1

1

112

121

2

)(

)(..;

10y

10y sCI

yyy1dt

dy

ydt

dy

2

1

1221

2

21

)(

)(..

)(

Stiff ODE – van del Pol equationStiff ODE – van del Pol equation

M-file for van del Pol equation

function yp = vanderpol(t,y,mu)

% van der Pol equation for electronic circuit

% d^2y1/dt^2 - mu*(1-y1^2) * dy1/dt + y1 = 0

% initial conditions, y1(0) = dy1/dt = 1

% convert to two first-order ODEs

% The solution becomes progressively stiffer

% as mu gets large

yp = [y(2); mu*(1-y(1)^2)*y(2)-y(1)];

Nonstiff ODE – van del Pol equationNonstiff ODE – van del Pol equation

= 1

>> [t,y]=ode45(@vanderpol,[0 20],[1 1],[], 1);

>> H=plot(t,y(:,1),t,y(:,2),'m--');

>> legend('y1','y2',2);

ode45

Stiff ODE – van del Pol equationStiff ODE – van del Pol equation

= 1000

>> [t,y]=ode23s(@vanderpol,[0 6000],[1 1],[], 1000);

>> H=plot(t,y(:,1));

>> set(H,'LineWidth',3)

ode23s

Bungee Jumper – Coupled ODEsBungee Jumper – Coupled ODEs

Vertical Dynamics of a jumper connected to a stationary platform with a bungee cord (Ex21.4)

Air resistance depending on whether the cord is slack or stretched – use sign(v) for drag force

Spring constant k (N/m) and damping coefficient (N·s/m)

Need to solve two simultaneous ODEs for x and v

vm

Lxm

kv

m

cvsigng

dt

dv 2d )()(

vm

Lxm

kv

m

cvsigng

dt

dv

vdt

dx

2d )()(

k = = 0 if x L

Non-stiff ODEs: Bungee JumperNon-stiff ODEs: Bungee Jumper

Use ode45 for non-stiff ODEs to solve for the distance and velocity of a bungee jumper

function dydt = bungee(t,y,L,cd,m,k,gamma)

g = 9.81;

cord = 0;

if y(1) > L % determine if the cord exerts a force

cord = k/m * (y(1)-L) + gamma/m * y(2);

end

dydt = [y(2); g - sign(y(2)) * cd/m*y(2)^2 - cord];

>> [t,y] = ode45(@bungee,[0 50],[0 0],[],30,0.25,68.1,40,8);

>> h1 = plot(t,-y(:,1),t,y(:,2),'m--');

>> h2 = legend('x (m)','v(m/s)');

>> set(h1,'LineWidth',3); set(h2,'FontSize',12);

Bungee jumper distance

Bungee jumper velocity

CVEN 302-501CVEN 302-501Homework No. 14Homework No. 14

Chapter 21Prob. 21.3 (30) (Hand Calculation) 21.5 (30)

Prob. 21.8 (40) (using MATLAB program except for part a))

Due 12/02/08 before 5:00 pmDue 12/02/08 before 5:00 pm