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Topic: highly accurate pde solvers?
Replies: 6   Last Post: Nov 30, 2012 5:34 AM

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Peter Spellucci

Posts: 218
Registered: 11/9/09
Re: highly accurate pde solvers?
Posted: Nov 29, 2012 9:15 AM
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Leslaw Bieniasz <nbbienia@cyf-kr.edu.pl> writes:
>
>
>Hi,
>
>I need to solve a certain relatively simple PDE
>(evolutionary convection-diffusion in 1D space)
>but with a very high accuracy, ideally with a relative error 10^(-20).
>Are there any free programs available that can handle such a problem?
>I presume the programs would have to use some sort of arbitrary precision
>arithmetics.
>
>Leslaw

here is a software which will help you:
this is from my annotations:
\begin{cite}
I am pleased to announce new extended precision arithmetic software. It
provides the following data types from either C or Fortran programs:

double-double 106 mantissa bits, or approx. 32 decimal digits
quad-double 212 mantissa bits, or approx. 64 decimal digits

In addition to the basic operations, numerous transcendental functions,
including sqrt, sin, cos and exp, are also included. This software is
slower than ordinary double precision, but is substantially faster than
using arbitrary precision packages for these precision levels.

This subroutine library is coupled with C++ and Fortran-90 translation
modules, which use operator and function name overloading, to
automatically perform the task of translating C or Fortran code to use
the library. Thus in most cases, only minor modifications need to made
to existing C or Fortran program to utilize these precision types. Here
is an example of a typical Fortran-90 program that uses the package:

program testdd
use qdmodule
type (qd_real) a, b, c
a = -1.d0
b = 2.d0
c = acos (a) * sqrt (b)
call qdwrite (6, c)
stop
end

This program produces the result
4.442882938158366247015880990060693698614621689375690223085395607E0
which is pi*sqrt(2) correct to 63 decimal digits.

This software was written by Yozo Hida of U.C. Berkeley, based on
earlier work by myself and Sherry Li of Lawrence Berkeley National
Laboratory. The software and documentation is available at the web site:

http://www.nersc.gov/~dhbailey/mpdist/mpdist.html

We would greatly appreciate any comments or bug reports by users of any
of this software. The above site also includes software for
multiprecision (arbitrary precision) arithmetic. The multiprecision
software will be revised for increased performance and usability in the
next few months.

DHB
David H Bailey "Computo ergo sum."
Lawrence Berkeley National Lab Tel. 510-495-2773
Mail Stop 50B-2239 Email: dhbailey@lbl.gov
\end{cite}

depending on the integrator you intend to use you will need 30 to 40
decimals precision , since of course the spatial and time steps you
can use will be quite small, for an order one method itself in the
range of 1.0e-20. higher order methods would help here very much.

hth
peter




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