> 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).
I suspect that such a level of accuracy isn't possible with single and double-precision floating point types, even if you manage to get the analytical solution to your PDE. Even the IEEE 754's double extended floating point data type is only able to express about 19 significant digits.
> 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.
I don't know of free programs, but I believe Mathematica does support arbitrary precision numbers.