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Science Forum Index » Math - Numerical Analysis Forum » high-precision numerical differentiation at a boundary
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| Charles Russell |
 Posted: Thu Mar 08, 2007 1:14 pm |
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What are the best high-order methods for numerical estimation of first
and second derivatives at a boundary, on one side of which the function
is undefined so that central differences are not available? |
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| Peter Spellucci |
| Posted: Fri Mar 09, 2007 5:53 am |
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In article <EuXHh.1312$nV1.1062@bignews6.bellsouth.net>,
Charles Russell <NOSPAM@bellsouth.net> writes:
Quote: What are the best high-order methods for numerical estimation of first
and second derivatives at a boundary, on one side of which the function
is undefined so that central differences are not available?
I assume you work with a finite difference boundary value solver
if your function (solution of an boundary value problem) cannot be extended
outside the domain (such that the trick of shifting the grid by stepsize/2
or using fictivious points cannot be used) then it is quite
questionable whether using high order difference approximations make
sense at all, but the usual choices here are a first or second order
finite difference with the boundary point as endpoint.
somthing like f'(x0)=(-3*f(x0)+4*f(x0+h)-f(x0+2*h))/(2*h)
why do you need
second order derivatives at the boundary? a fourth order problem?
better reformulated as coupled system of second order equations?
think about the conditioning of fourth order problems!
hth
peter |
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| Charles Russell |
| Posted: Fri Mar 09, 2007 2:40 pm |
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Peter Spellucci wrote:
Quote: In article <EuXHh.1312$nV1.1062@bignews6.bellsouth.net>,
Charles Russell <NOSPAM@bellsouth.net> writes:
What are the best high-order methods for numerical estimation of first
and second derivatives at a boundary, on one side of which the function
is undefined so that central differences are not available?
why do you need
second order derivatives at the boundary?
The question arose as follows: If you have convergence problems in an
optimization problem that depends on finite-difference derivatives and
want to see whether the situation is improved with more accurate
derivatives, how do you handle a boundary where the function becomes
undefined? What method is simplest, and what method is best?
I got my problem solved without resorting to higher-order derivatives,
but still remain curious about the question. |
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