Runge-kutta Methods

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Ryan Sullenberger 2 May 2007 ME102

Runge-Kutta Methods Fourth-Order RK Method 25.8 : Solve the following problem with the fourth-order RK method where y(0) = 4 and y`(0) = 0. Solve from x = 0 to 5 with h = 0.5. Plot the results.

DiffEQ and Initial Conditions:

d

2

dx

2

 d y  + 8⋅ y = 0   dx 

y + 0.6⋅ 

d y = y1 dx

d

2

dx

2

y = −0.6

d y − 8y = −0.6y 1 − 8y 0 dx

d y(0) = 0 dx

y( 0) = 4

System of first order ODE's and IC's:

4 Y0 :=   0

Y   1   D( x , Y) :=  −0.6⋅ Y1 − 8⋅ Y0   

4th-order RK solution: N := 100

xY := rkfixed ( Y0 , 0 , 5 , N , D)

(

xY = x Y

0

Y

1

)

0

xY =

1

2

0

0

4

0

1

0.05

3.96

-1.571

2

0.1

3.844

-3.065

3

0.15

3.656

-4.454

4

0.2

3.401

-5.714

5

0.25

3.087

-6.825

6

0.3

2.721

-7.769

7

0.35

2.313

-8.532

8

0.4

1.871

-9.105

9

0.45

1.406

-9.482

10

0.5

0.926

-9.661

11

0.55

0.443

-9.644

12

0.6

-0.035

-9.439

13

0.65

-0.498

-9.054

14

0.7

-0.938

-8.502

15

0.75

-1.346

...

〈0〉 X := xY 〈1〉 S1 := xY 〈2〉 S2 := xY

Graphed solution: 10

5

S1 S2

0

−5

− 10

0

1

2

3 X

4

5

Solving the ODE by writing a program that engages fourth-order RK method: h := 0.5

N := 10

f1( z) := z

f2( y , z) := −0.6⋅ z − 8 ⋅ y

( x y z ) :=

y ←4 0

z ←0 0

x ←0 0

for i ∈ 0 .. N − 1 x

i+ 1

← (i + 1)⋅ h

( i) k12 ← f2( y , z ) i i

k11 ← f1 z

y_new ← y +

1

z_new ← z +

1

i

i

2 2

⋅ k11 ⋅ h ⋅ k12 ⋅ h

k21 ← f1( z_new) k22 ← f2( y_new , z_new) y_new ← y +

1

z_new ← z +

1

i

i

2 2

⋅ k21 ⋅ h ⋅ k22 ⋅ h

k31 ← f1( z_new) k32 ← f2( y_new , z_new) y_new ← y + k31 ⋅ h i

z_new ← z + k32 ⋅ h i

k41 ← f1( z_new) k42 ← f2( y_new , z_new) y z

i+ 1

i+ 1

←y +

1

←z +

1

i

(x y z )

i

6 6

⋅ ( k11 + 2 ⋅ k21 + 2 ⋅ k31 + k41 ) ⋅ h ⋅ ( k12 + 2 ⋅ k22 + 2 ⋅ k32 + k42 ) ⋅ h

Programs solutions: 0

x=

0

0

0

0

0

4

0

0

1

0.5

1

1.037

1

-9.289

2

1

2

-2.428

2

-3.197

3

1.5

3

-1.557

3

5.365

4

2

4

1.154

4

4.072

5

2.5

5

1.481

5

-2.333

6

3

6

-0.293

6

-3.638

7

3.5

7

-1.132

7

0.372

8

4

8

-0.185

8

2.66

9

4.5

9

0.724

9

0.656

10

5

10

0.378

10

-1.626

y=

z=

Compare to MathCAD's rkfixed solutions:

Programs solutions graphed:

10

10 y

5 y z

5

z 0

S1 S2

−5 − 10

0

1

2

3 x

4

5

0 −5 − 10

0

1

2

3

x, x, X, X

4

5

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