Scientific Computing LabResults Worksheet 2

Dr. Miriam Mehl

Institut fürInformatik

Scientific Computingin

Computer Science

Worksheet 2 – Solution

Worksheet 2 – Solution

1) m-file Euler_implicit.m

for (k=2:size(tk,2)) y_it = y(k-1); dy=1.0; %Newton iteration while (abs(dy)>=0.0001)

dy = (y(k-1) - y_it + dt*rhs(y_it))/ (dt*rhs_y(y_it)-1);

y_it = y_it - dy; end y(k)=y_it;

end

Worksheet 2 – Solution

2) m-file Adams_Moulton.m

y_it = y(k-1); x = y_it + (1/2)*dt*rhs(y_it); %Newton iteration d=1-7*dt+(49/4)*dt^2+(7/5)*dt*x; if (d>0)

dy=1; it=1; while (abs(dy)>0.0001) …

Worksheet 2 – Solution

3) m-file AdamsMoulton_linear_1.m

for (k=2:size(tk,2)) a=(1/2)*dt*7*y(k-1); y(k)=(y(k-1)+2*a-(a/10)*y(k-1))/

(1+(a/10)); end

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error – – 2.0595 0.4854 0.1754

error red. – – – 4.24 2.77

explicit Euler

q=1

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error – – 0.0950 0.0938 0.0235

error red. – – – 1.01 3.99

Heun

q=2

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error 0.5781 0.5151 0.3665 0.2233 0.1245

error red. – 1.12 1.41 1.64 1.79

implicit Euler

q=1

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error – 1.4730 0.3035 0.0704 0.0167

error red. – – 4.85 4.31 4.21

Adams Moulton

q=2

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error 1.8196 0.8402 0.3342 0.1237 0.0488

error red. – 2.17 2.51 2.69 2.53

Adams Moulton linearisation 1

q=1

Worksheet 2 – Solution

dt 1/2 1/4 1/8 1/16 1/32

error 18.856 2.1543 0.7990 0.2326 0.1220

error red. – 8.75 2.70 2.73 2.40

Adams Moulton linearisation 2

q=1

Worksheet 2 – Solution

dt Expl. Euler

Heun Impl. Euler

Adams Moulton

Adams Moulton

l1

Adams Moulton

l2

1/2 X – X O

1/4 X X X X

1/8 X X X X X X

1/16 X X X X X X

1/32 X X X X X X