The Computable Differential Equation Lecture ... - Bruce E. Shapiro

82 CHAPTER 4. IMPROVING ON EULER’S METHOD
We could implement this in Mathematica as follows:
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BackwardEulerFP[f , {t0 , y0 }, h , tmax , tol , imax ] :=
Module[{ time, yval, yvaln, yvalp, delta, eps, r},
r = {{t0, y0}}; yval = y0;
For[time = t0, time < tmax, time += h,
(***** Do next Backward Euler Step *****)
yvaln = yval + f[time+h, yval];
(***** Do Fixed Point Iteration on y *****)
For[i = 1, i ≤ imax, i++,
yvalp = yvaln;
yvaln = yval + f[time+h, yvalp];
delta = yvaln - yvalp;
If[Abs[delta] < tol, Break[]];
];
yval = yvaln;
AppendTo[r, {time + h, yval}]; ];
Return[r]; ]
We know that equation 4.103 is only guaranteed to have a fixed point, and iteration
is only guaranteed to converge, if it is a contraction mapping. This requires
‖hf y (t, y)‖ < 1. Convergence becomes rapid when ‖hf y (t, y)‖ y0, t -> t0;
If[Abs[JV] >= 1,
Print["This problem is ill-conditioned for fixed point iteration."];
Return[$Failed];
];
This calculates the scalar Jacobian analytically, and places the formula in the
variable J , and then calculates the numerical value of the Jacobian in the variable
JV . If the magnitude of the Jacobian is larger than or equal to one, then fixed point
iteration is not guaranteed to work, so the program aborts. Later in the algorithm,
after the next step is calculated, one also needs to test to see if the problem has
entered an ill-conditioned region. To do this one can recalculate the value of the
Jacobian, using the analytic formula already determined, merely substituting the
current values of y and t.
JV = J/.y-> ynext, t-> time
If[Abs[JV]>= 1 ...
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007