Re: higher order finite differencing scheme

From: Peter Spellucci (spellucci_at_fb04373.mathematik.tu-darmstadt.de)
Date: 12/20/04 

Date: Mon, 20 Dec 2004 10:49:33 +0000 (UTC)

In article <cpsh30$sag$1@dsnews.hmi.de>,
 Benedikt Obermayer <obermayer@hmi.de> writes:
>Hi all,
>
>I'm writing a solver for a nonlinear two-point boundary value problem
>(something between the relaxation solver of numerical recipes and the
>ones used e.g. by IMSL or NAG libraries). I'm trying to improve my
>results (implementing deferred corrections) by using higher order
>differencing schemes, i.e. instead of approximating u'(x)=f(x,u(x)) by
>
>u'(x_{i+1/2})= ( u(x_{i+1})-u(x_i) )/h + O (h^2)
>
>something that couples more than just two points. Now I've found some
>literature on these things, some are using mono-implicit Runge-Kutta
>formulae, others try to approximate the error term which involves u" by
>the derivative f' of the RHS. My question is: why not something like
>
>u'(x_{i+1/2})=a u(x_{i-1})+b u(x_i)+c u(x_{i+1})+d u(x_{i+2})+O(h^4)
>
>and getting a,b,c,d again from Taylor-Expansion with the required
>changes near the boundaries.
>
>Of course there are a few obvious counter-arguments:
>- I would be assuming that u is quite smooth, but it probably is
>(actually, it's a second order equation transformed into a system), and
>for me as a physicist that's not the main point
>- the linear system to be solved has still band-structure, but the
>bandwidth is greater; but, performance is not the main issue here and
>the complexity is the same if one is using specialized solvers (or not?)
>
>I'm asking because the evaluation of my RHS is very expensive, so I
>don't want to do the Runge-Kutta. And it seems to be more
>straightforward. Is there any practical reason (stability, convergence)
>not to use "my" idea or just because of the smoothness assumptions?
>
>Thanks for answers,
>
>Benedikt

you write nothing about the system you are considering and since you are using
only u' I assume it is
   u' =F(t,u), R(u(a),u(b))=0.
the first approximation you are mentioning is the simplest collocation scheme.
Using finite differences of higher order will cause considerable trouble
in the nonlinear system to be solved, including the trouble of special treatment
of grid points near the boundary. you will also have to use a fourth order
approximation of u(x(i+1/2)) inside F and this in turn will influence the
properties of the Jacobian of the nonlinear system to be solved.
therefore I would prefer to use the simple second order scheme with several
stepsizes and an extrapolation ala Richardson/Romberg
hth
peter