Re: Working on bvp4c

[lakerluke]

Hi Carlo,

I have been doing some investigation into bvp4c and would like to ask you
some questions relating to the Shampine paper you linked. I believe that the
first step in implementing bvp4c, in the manner of the algorithm outlined in
the Shampine paper, is to form the residual approximation that appears at
the middle of page 5 (I will refer to this equation as [1]). Then, the
integral norm must be taken of this residual in order to base error
estimation and mesh selection.

The equation for the residual [1] depends on the 4th derivative of the
solution y(x). My question is how best to form this term? Since dy/dx = f(t,
y) we can use the chain rule and take continuous derivatives of this
governing equation to form the 4th derivative of y in terms of f and
derivatives of f. However, this will need recalculating at each iteration
and I don't believe this is the best method to use.

The Shampine paper goes on to explain on page 6 that we can apply the
Simpson method to system (4.1) and further explains some simplification that
can be done for the Jacobian terms. However, applying the Simpson method
results in a cubic spline and hence if this is used as the approximation to
y(x) then upon taking the 4th derivative of the spline we will get a
constant function of zero at all points?

Finally, at the top of page 9, the paper expresses the residual in terms of
the solution to the system (4.1), (I will refer to this expression of the
residual at the midpoint as [2]). Since the residual at the midpoint is an
indication of how well our equations were satisfied, then this expression
[2] is the residual we wish to minimize. It then goes on to say that [2] is
used by the quadrature formula for the norm of the residual (i.e. [1]
mentioned earlier). I fail to see how [2] can be applied to [1] since they
are evaluated at different points.

Apologies for the length email but I'm hoping someone could help me out with
my questions:

1) How to best form the term for the 4th derivative of the solution
evaluated at the midpoint in order to form [1]?

2) How expression [2] is used in the evaluation of [1]?

Best Regards,

Luke



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