Re: Working on bvp4c

[c.]

On 3 Aug 2016, at 08:53, lakerluke <address@hidden> wrote:

> 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

Hi Luke,

Thanks for working on this!

I am traveling and I am unfortunately not able 
to give a detailed answer right now, will try to 
do so when I am back at the end of next week.

Anyway I wanted to give a quick hint as I am afraid
you might spend too much time workin in the wrong direction...

One thing you have to keep in mind is that 
you will never need to compute any high order
derivatives, this is an important feature of the
method.

I can only guess as I don't have the paper at hand
right now, but I am quite sure that, if the formula 
you refer to includes a fourth-order derivative, than 
that formula is meant for the (theoretical) analysis 
of the error, it is not an expression to be used in
the implementation.

I'll come back to this as soon as I'm back,
c.