Lecture 34 - Ordinary Differential Lecture 34 - Ordinary Differential Equations - BVP Equations - BVP

CVEN 302

November 28, 2001

Systems of Ordinary Differential Systems of Ordinary Differential Equations - BVPEquations - BVP

• Shooting Method for Nonlinear BVP

• Finite Difference Method

• Partial Differential Equations

Shooting Method for Nonlinear Shooting Method for Nonlinear ODE-BVPsODE-BVPs

• Nonlinear

ODE

• Consider with guessed slope t

• Use the difference between u(b) and yb to adjust u’(a)

• m(t) = u(b, t) - yb is a function of the guessed value t

• Use secant method or Newton method to find the correct t value with m(t) = 0

ba y)b(y ,y)a(ybxa , )y,y,x(fy

t)a(u ,yu(a) )u,u,x(fu

a

Nonlinear Shooting Based on Secant Nonlinear Shooting Based on Secant MethodMethod

• Nonlinear

ODE

0))b(y),b(yh( ,y)a(ybxa , )y,y,x(fy

a

tol1)-t(i-t(i) until iterate :4 Step

1)m(i2)m(i1)m(i

2)t(i1)t(i1)t(it(i)

estimate new a obtain to methodsecant use :3 Step

m(2)Error t(2)(a)u ,yu(a) use:2 Step

m(1)Error t(1)(a)u ,yu(a) use:1 Step

a

a

MATLAB Example in Nonlinear Shooting MATLAB Example in Nonlinear Shooting MethodMethod

• Nonlinear shooting with secant method

• Convert to two first-order ODE-IVPs

• Update t using the secant method

)1x/(1y solution exact

025.0)1(y)1y())1(y),1(y( , h1)0y(

1x0 , yy2y

t)0(z ,zz2z

1)0(z , zz

yz ,yz let

2212

121

21

Nonlinear Shooting - Secant MethodNonlinear Shooting - Secant Method

y(x)

y’(x)

Nonlinear ShootingNonlinear Shooting Based on Newton’s Method Based on Newton’s Method

• Nonlinear

ODE

Check for convergence of m(t)

ba y)b(y ,y)a(y

bxa , )y,y,x(fy

1(a)v 0,v(a)

t(a)u ,yu(a)

)uu,(x,fv )uu,(x,fvv

)uu,f(x,u ka

uu

)t,b(v

mtt ,Otherwise

stop ,tolm if

y)tu(b,)tm(

kk1k

bkk

Nonlinear ShootingNonlinear Shooting Based on Newton’s Method Based on Newton’s Method

• Nonlinear

ODE-IVP

• Chain Rule

b

a

y)t,b(u)t(m

t)a(u ,y)a(u , )u,u,x(fu

t

u

u

f

t

u

u

f

t

x

x

f

t

)u,u,x(f

t

u

1t/u(a)v' t)t,a(u

vfvfvt

u

u

f

t

u

u

f

t

ut

uv ,

t

uv then ,

t

uv let

uu

x and t are independent

0

Nonlinear Shooting with Newton’s MethodNonlinear Shooting with Newton’s Method

• Solve ODE-IVP

• Construct the auxiliary equations

1)0(u, 1)0u(

, u

)u(u

2)1(, y1)0y(

, y

)y(y

22

1)0(v ,0)0(v

vu

u2v

u

)u(v

u

u2

u

f)u,u,x(f

u

)u(

u

f)u,u,x(f

2

2

u

2

2

u

Nonlinear Shooting with Newton’s MethodNonlinear Shooting with Newton’s Method

• Calculate m(t) -- deviation from the exact BC

• Update t by Newton’s method

),()(

),()(

tbutm

ytbutm

tt

b

),( ;)(

)(tbu

t

m(t)m v

tm

tmtt tt

1it

1i1ii

Finite-Difference MethodsFinite-Difference Methods

• Divide the interval of interest into subintervals

• Replace the derivatives by appropriate finite-difference approximations in Chapter 11

• Solve the system of algebraic equations by methods in Chapters 3 and 4

• For nonlinear ODEs, methods in Chapter 5 may be used

i1in0 xxh

abh ,bx ,ax

Finite-Difference MethodFinite-Difference Method

• General Two-Point BVPs

• Replace the derivatives by appropriate finite-difference approximations

y(b) ,y(a)

bxa ),x(ry)x(qy)x(py

h2

yy)x(y ,

h

yy2y)x(y 1i1i

i21ii1i

i

xixi-1 xi+1

hh h h

Finite-Difference MethodFinite-Difference Method

• Central difference approximations

• Tridiagonal system

i2

1iiii2

1ii

iii1i1i

i21ii1i

rhy p2

h1y qh2y p

2

h1

ryqh2

yyp

h

yy2y

21ii1i

i

1i1ii

h

yy2y)x(y

h2

yy)x(y

Finite-Difference MethodFinite-Difference Method

• Central Difference ==> Tridiagonal system

)/2ph1(rh

rh

rh

)/2ph1(rh

y)/2ph1(rh

rh

rh

y)/2ph1(rh

y

y

y

y

)qh2(000

0)qh2(p(h/2)10

0p(h/2)1)qh2(p(h/2)1

000)qh2(

1-n1n2

32

22

112

n1-n1n2

32

22

0112

1n

3

2

1

1n2

32

3

222

2

12

Finite-Difference Method for Finite-Difference Method for Nonlinear BVPsNonlinear BVPs

• Nonlinear ODE-BVPs

• Evaluate fi by appropriate finite-difference approximations

y(b) ,y(a)

bxa , )y,y,x(fy

0fh

yy2yi2

1ii1i

xixi-1 xi+1

hh h h

Finite-Difference Method for Finite-Difference Method for Nonlinear BVPsNonlinear BVPs

• SOR method

• Iterative solution• Convergence criterion

)fhyy 2y()1(2

1y i

21ii1ii

P)y,y,x(f

Q)'y,y,x(fQ0 with

P/2h

2/Qh

y

y2

Example 14.12 - MATLABExample 14.12 - MATLAB

• Note error in Text

i

2

21i1i

i

21i1i

i

i2

1ii1ii

yh4

)yy(

y

h2/)yy( f

)fhyy 2y()1(2

1y

2 y(1) , 1 y(0) ,y

)y()y,y,x(fy

2

fi : negative sign

Chapter 15

Partial Differential Equations

Classification of PDEsClassification of PDEs• General form of linear second-order PDEs with

two independent variables

• linear PDEs: a, b, c,….,g = f(x,y) only

0gfueuducubuau yxyyxyxx

roots)complex (2 Elliptic ,0ac4b

root)double (1 Parabolic ,0ac4b

roots) real (2 Hyperbolic ,0ac4b

2

2

2

Heat Equation: Parabolic PDEHeat Equation: Parabolic PDE

• Heat transfer in a one-dimensional rod

Tt0 , )t(g)t,a(u

)t(gt)u(0,

ax0 f(x),u(x,0)

Tt0 a,x0 ;x

uc

t

u

2

1

2

2

x = 0 x = ag1(t) g2(t)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Discretize the solution domain in space and time with h = x and k = t

Time(j index)

space (i index)

x

t

Initial and Boundary ConditionsInitial and Boundary Conditions

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Initial conditions : u(x,0) = f(x)

u(0, t) = g1(t)

u(a, t) = g2(t)

Explicit Euler method

Heat EquationHeat Equation• Finite-difference

(i,j) (i+1,j)(i-1,j)

(i,j+1)u(x,t)

xx

t

t

xi xi+1xi-1

tj

tj+1

)uu2u(h

ccu

)uu(k

1u

j,1ij,ij,1i2xx

j,i1j,it

Forward-difference

Central-difference

at time level j

Explicit MethodExplicit Method

• Explicit Euler method for heat equation

• Rearrange

jkt ,m/Tt kihx ,n/ax h

letj

i

)uu2u(h

c)uu(

k

1cuu j,1ij,ij,1i2j,i1j,ixxt

j,1ij,ij,1i

j,1ij,ij,1i2j,i1j,i

ruu)r21(ru

)uu2u(h

ckuu

22 x

t c

h

ckr

5.0r0 Stability:

Explicit Euler MethodExplicit Euler Method

• Stable

• Unstable (negative coefficients)

j,1ij,ij,1i1j,i ruu)r21(ruu

j,1ij,1i1j,i

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

u 5.0u 5.0u 5.0r

u 4.0u 2.0u 4.0u 4.0r

u 1.0u 8.0u 1.0u 1.0r

u 01.0u 98.0u 01.0u 01.0r

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

j,1ij,ij,1i1j,i

u100u 199u100u 100ru 10u 19u 10u 10r

u u u u 1r

Heat Equation: Explicit Euler MethodHeat Equation: Explicit Euler Method

r = 0.5

Example: Explicit Euler MethodExample: Explicit Euler Method • Heat Equation (Parabolic PDE)

• c = 0.5, h = 0.25, k = 0.05

e60)t,1(u ,e20t)u(0,

x4002u(x,0)

1x0 ;cuu

t2t

xxt

1 2 3 4020 + 40 x

60e -2t20e -t

0

1

2

ExampleExample • Explicit Euler method

• First step: t = 0.05

j,1ij,ij,1i

j,1ij,ij,1i1j,i

22

u4.0u2.0u4.0

ruu)r21(ruu

4.0)25.0(

)05.0)(5.0(

h

ckr

29024508.54e60u

50)60(4.0)50(2.0)40(4.0u4.0u2.0u4.0u

40)50(4.0)40(2.0)30(4.0u4.0u2.0u4.0u

30)40(4.0)30(2.0)20(4.0u4.0u2.0u4.0u

02458849.19e20u

10.01,4

0,40,30,21,3

0,30,20,11,2

0,20,10,01,1

05.01,0

• Second step: t = 0.10

1 2 3 4020 + 40 x

60e -2t20e -t

12384518.49e60u

71609803.47)2924508.54(4.0)50(2.0)40(4.0

u4.0u2.0u4.0u

40)50(4.0)40(2.0)30(4.0u4.0u2.0u4.0u

6098354.29)40(4.0)30(2.0)02458849.19(4.0

u4.0u2.0u4.0u

09674836.18e20u

20.02,4

1,41,31,22,3

1,31,21,12,2

1,21,11,02,1

10.02,0

30 40 50

29.61 40 47.72

Heat Equation: Time-dependent BCsHeat Equation: Time-dependent BCs

r = 0.4

• Stability for Explicit Euler Method

• It can be shown by Von Neumann analysis that

• Switch to Implicit method to avoid instabilityc

x

2

1t or

2

1r

2

Numerical StabilityNumerical Stability

Explicit Euler Method: StabilityExplicit Euler Method: Stability

Unstable !!

r = 1

Implicit Euler methodImplicit Euler method

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Initial conditions : u(x,0) = f(x)

u(0, t) = g1(t)

u(a, t) = g2(t)

Unconditionally Stable

Implicit MethodImplicit Method• Finite-difference

(i,j)

(i+1,j+1)(i-1,j+1) (i,j+1)

T(x,t)

xx

t

t

xi xi+1xi-1

tj

tj+1

)uu2u(h

ccu

)uu(k

1u

1j,1i1j,i1j,1i2xx

j,i1j,it

Forward-difference

Central-difference at time level j+1

Implicit Euler MethodImplicit Euler Method • Implicit Euler method for heat equation

• Tridiagonal matrix (Thomas algorithm)

• Unconditionally stable

j,i1j,1i1j,i1j,1i

1j,1i1j,i1j,1i2j,i1j,i

uruu)r21(ru

)uu2u(h

c)uu(

k

1

1j,nj,1n

j,3

j,2

1j,0j,1

1j,1n

1j,3

1j,2

1j,1

ruu

u

u

ruu

u

u

u

u

r21r

r

rr21r

rr21r

rr21

Implicit Euler MethodImplicit Euler Method

Unconditionally stable

r = 2

Example: Implicit Euler MethodExample: Implicit Euler Method • Heat Equation (Parabolic PDE)

• c = 0.5, h = 0.25, k = 0.1

e60)t,1(u ,e20t)u(0,

x4002u(x,0)

1x0 ;cuu

t2t

xxt

1 2 3 4020 + 40 x

60e -2t20e -t

0

1

ExampleExample • Implicit Euler method

j,i1j,1i1j,i1j,1i

j,i1j,1i1j,i1j,1i

22

uu)8.0(u)6.2(u)8.0(

uu)r(u)r21(u)r(

8.0)25.0(

)10.0)(5.0(

h

ckr

1403

02

1001

13

12

11

ruu

u

ruu

u

u

u

r21r0

rr21r

0rr21

,,

,

,,

,

,

,

• Solve the tridiagonal matrix

1 2 3 4020 + 40 x

60e -2t20e -t

0

1

19426454.46

50751457.38

95515793.28

u

u

u

)e60(8.050

40

)e20(8.030

u

u

u

6.28.00

8.06.28.0

08.06.2

1,3

1,2

1,1

2.0

1.0

1,3

1,2

1,1

28.96 38.51 46.19

Crank-Nicolson methodCrank-Nicolson method

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

Initial conditions : u(x,0) = f(x)

u(0, t) = g1(t)

u(a, t) = g2(t)

Implicit Euler method : first-order in timeCrank-Nicolson : second-order in time

Crank-Nicolson MethodCrank-Nicolson Method • Crank-Nicolson method for heat equation• Average between two time levels

• Tridiagonal matrix

• Unconditionally stable (neutrally stable)• Oscillation may occur

)uu2u(h2

c

)uu2u(h2

c)uu(

k

1

1j,1i1j,i1j,1i2

j,1ij,ij,1i2j,i1j,i

j,1ij,ij,1i1j,1i1j,i1j,1i u2

ru)r1(u

2

ru

2

ru)r1(u

2

r

General Two-Level MethodGeneral Two-Level Method • General two-stage method for heat equation

• Weighted-average of spatial derivatives between two time levels n and n+1

)uu2u(h

)1(c

)uu2u(h

c)uu(

k

1

1j,1i1j,i1j,1i2

j,1ij,ij,1i2j,i1j,i

scheme Nicolson-Crank :1/2λscheme Euler explicit :1λscheme Euler implicit :0λ

Example: Crank-Nicolson MethodExample: Crank-Nicolson Method • Heat Equation (Parabolic PDE)

• c = 0.5, h = 0.25, k = 0.1

e60)t,1(u ,e20t)u(0,

x4002u(x,0)

1x0 ;cuu

t2t

xxt

1 2 3 4020 + 40 x

60e -2t20e -t

0

1

ExampleExample • Crank-Nicolson method

• Tridiagonal matrix (r = 0.8)

j1,iji,j1,i1j1,i1ji,1j1,i

j1,iji,j1,i1j1,i1ji,1j1,i

22

0.4u0.2u0.4u0.4u1.8u0.4u

u2

rr)u(1u

2

ru

2

rr)u(1u

2

r

8.0)25.0(

)10.0)(5.0(

h

ckr

1,40,40,30,2

0,30,20,1

1,00,20,10,0

1,3

1,2

1,1

u2

ru

2

ru)r1(u

2

r

u2

ru)r1(u

2

r

u2

ru

2

ru)r1(u

2

r

u

u

u

r12

r0

2

rr1

2

r

02

rr1

• Solve the tridiagonal matrix

1 2 3 4020 + 40 x

60e -2t20e -t

0

1

42746748.47

29975855.39

42144598.29

u

u

u

64953807.69

40

23869934.37

)e60(4.0)60(4.0)50(2.0)40(4.0

)50(4.0)40(2.0)30(4.0

)e20(4.0)40(4.0)30(2.0)20(4.0

u

u

u

8.14.00

4.08.14.0

04.08.1

1,3

1,2

1,1

2.0

1.0

1,3

1,2

1,1

29.42 39.30 47.43

Implicit Euler methodImplicit Euler method

Unconditionally stable

r = 2

Heat Equation with Insulated Heat Equation with Insulated BoundaryBoundary

• No heat flux at x = 0 and x = a

Tt0 , 0)t,a(u

0t)(0,u

ax0 f(x),u(x,0)

Tt0 a,x0 ;x

uc

t

u

x

x

2

2

x = 0 x = a

ux(a,t) = 0ux(0,t) = 0

Insulated BoundaryInsulated Boundary• No heat flux at x = a

1n1n1n1n

j,1nj,1nx uu 0

xx

uu)t,a(u

j,nj,1n

j,1nj,nj,1n1j,n

u)r21(ru2

ruu)r21(ruu

x = a

xn+1xn-1 xn

ux(a,t)=0