I need to solve a certain relatively simple PDE (evolutionary convection-diffusion in one space dimension) but the problem is that I need a very accurate solution. Ideally I would like to have 20 significant digits, less ideally 16 digits, and in the worst case 10-12. Are there any freely available procedures for MATHEMATICA that can do this job? I presume some sort of adaptive solvers would be needed. I notice that NDSolve can realistically provide no more than 4 significant digits, even if one plays with AccuracyGoal and PrecisionGoal parameters, which do not seem to have much effect on the number of digits obtained. What is worse, if one selects too many spatial nodes, to satisfy these error tolerances, the program crashes due to insuffucient memory.