Implementing a Newton-Raphson iteration method

Question

I'm trying to implement a backward Euler Scheme using the Newton Raphson iteration. I understand that with each iteration, one makes an initial guess, calculates the residual and solve for the change. In my case the change is del w. After that I know to add the value to w^m and get an update value for w at the next m iteration. I know to check the convergence of the solution as the iterations go on. The problem I am having is how to implement the time step dt as t=0:Tmax/dt where Tmax is say 10. I'm confused on how the time stepping comes in. I've been trying to figure this out for a while so any help will be appreciated. Thank you!

 while Rw(m)>10^-6      % Convergence condition
    drdw(m)=(1-2*dt+2*t(n+1)^2*w(m)*dt);
    Dw(m)=Rw(m)\drdw(m); %Inverse
    w(m+1)=w(m)+Dw(m);  %Newton method
    Rw(m+1)=(-(w(m)-v(1)-2*w(m)*dt+t(n+1)^2*w(m)^2*dt));   %New Residual value
    if Rw(m+1)>10^-6  %Check on the level of convergence
        m=m+1;
    else
        Rw=1;  % I was thinking I should make the Residual 1 for the next time step. 
        break

    end

phwd phwd · Accepted Answer · 2011-02-25T01:17:46

Something is strange in there, a 1st order ODE is of the form dy/dt = f(t,y) yet you have w,v,t,m and n.

You are computing your solution over the time interval (0,T) into a partition with constant step size h:

t0 = 0;      tk = hk;      tn = T

For this

Backward Euler

Try implementing your newton method (using the while conditional as you have above) to solving the above as a separate function and then integrate it into your euler over the steps

1 to n where n = Tmax/h.

Implementing a Newton-Raphson iteration method

1 Answers