Approximate solution for spring mass spring damper using backward (implicit),improved euler (predictir-corrector), central difference, and runge-kutta

Hi guys,

Im trying to solve the response for mass-spring-damper system using the following approximate methods:

Euler (backward difference)

improved euler (predictor-corrector)

Central difference

Runge-Kutta

I already developed code that does forward difference but i dont know how to modify it to cover the rest that is mentioned above.

I would appriciate it if you could helo me with above methods. the equation of motion for the mentiioned system is:

and my code is attached

dz1dt=@(t,z1,z2) (z2); %coverting 2nd degree ode into 2 1st order 
dz2dt=@(t,z1,z2) (-2*Z*omegan*z2-omegan^2*z1);
z1_0=1; %initial condition
z2_0=0;
a=0; %time start
b=10;
h=0.01; %step size 
z1i=z1_0;
z2i=z2_0;
ti=a;
iter=1;
zout(1,1:2)=[z1i,z2i];
tout(1)=a;
while ti<b;
    z1ip1=z1i+h*dz1dt(ti,z1i,z2i) %solving used forwarde diff
    z2ip1=z2i+h*dz2dt(ti,z1i,z2i)
    tip1=a+iter*h;
    zout(iter+1,1:2)=[z1i,z2i];
    tout(iter+1,1)=tip1;
    iter=iter+1;
    ti=tip1;
    z1i=z1ip1;
    z2i=z2ip1;
end
figure()
    plot(tout,zout(:,1))

[darova]

• improved euler (predictor-corrector)

Do you have any attempts on this? What kind of help do you need?

[Faraz Vossoughian]

Im not sure how to modify forward to predicor-corrector

[darova]

Do you have formulas?

[Faraz Vossoughian]

yes some of the formulas are attached in the pdf below