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Re: highly accurate pde solvers?
Posted:
Nov 29, 2012 9:15 AM


Leslaw Bieniasz <nbbienia@cyfkr.edu.pl> writes: > > >Hi, > >I need to solve a certain relatively simple PDE >(evolutionary convectiondiffusion 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:
doubledouble 106 mantissa bits, or approx. 32 decimal digits quaddouble 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 Fortran90 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 Fortran90 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. 5104952773 Mail Stop 50B2239 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 1.0e20. higher order methods would help here very much.
hth peter



