Re: 2nd derivative FD discretisations


In article <Pine.GHP.4.58.0511181250001.6346@xxxxxxxxxxxxxxxxxxx>,
Leslaw Bieniasz <nbbienia@xxxxxxxxxxxxx> writes:
>
> Cracow, 18.11.2005
>
>Hi,
>
>The classical 3-point finite-difference discretisation of a
>second derivative of function u(x) on a nonuniform grid of
>points x-, x0, x+ is
>
>u''(x0) =
>
> 2 {[u(x+) - u(x0)]/(x+ - x0) - [u(x0) - u(x-)]/(x0 - x-)}/ (h- + h+)
>
>and has a truncation error with a dominant term
>(1/3) u'''(x0) [(x+ - x0) - (x0 - x-)]
>
>which means that it is only first order accurate, in contrast to the
>same scheme on a uniform grid, which is second-order accurate.
>
>Do there exist any alternative classical (non-Hermitian, non-compact)
>three-point schemes for 2nd derivative that would be 2nd order accurate
>also on non-uniform grids?
>
>I need the answer in the context of solving BVPs for ODEs
>of the type
>
>F(x, u, u', u'') = 0
>
>or
>
>u'' = G(x, u, u')
>
>Any help will be appreciated.
>
>L.B.
>
>
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>| Institute of Physical Chemistry of the Polish Academy of Sciences,|
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no. if you write down the general 3 point finite difference using taylors
theorem then you see that it is impossible to get second order accuracy without
the symmetry, because then you are unable to eliminate the third derivative.
there is at the first glance a trick: use a first order system introducing
v1=u, v2=u' and now approximate v1' and v2' with a general three point
scheme of second order. but this finally results in a five band jacobian
again if you number the unknowns consecutively, hence nothing is won.
(the matrix will be better conditioned though, but indefinite)
hth
peter
.

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