I think that notation might have me fooled.
At each step, and are constants and so,
This means that,
at each step in order to be sure that fixed-point iteration will converge.
the trapezoid method of solving differential equations is,
Since the method is implicit, we must solve for on each step by some iterative method. If we use the fixed point iteration , we know that the method will converge if for .
Give a condition on such that the fixed point iteration converges at each step.
First of all, I need to write down an expression for . Fixed-point iteration needs an initial guess. I use as my initial guess and write as,
which doesn't make much sense.
What am I doing wrong?