In the vast majority of cases, people using ODE solvers are working with physical systems, where there likely aren't more than a dozen digits of accuracy in any of the parameters. So you set some fixed level of accuracy for the output, and seek a method that can reliably attain that accuracy at the lowest cost. (Either that, or the system is too chaotic, in which case you're told not to bother since the output would be meaningless anyway.)
Something I've had trouble finding any information about is, what if we have a mathematical system, where the initial configuration is known to infinite precision, and we want to solve the ODE to an arbitrary precision (say, 100 digits, or 1000 digits)? All the usual stepwise methods would seemingly require a very high order to avoid a galactic step count, but initializing all the coefficients, etc., would be a struggle in itself.
Is this the best that can be done, or are there less-usual methods that would win out at extremely high accuracy levels?
Something I've had trouble finding any information about is, what if we have a mathematical system, where the initial configuration is known to infinite precision, and we want to solve the ODE to an arbitrary precision (say, 100 digits, or 1000 digits)? All the usual stepwise methods would seemingly require a very high order to avoid a galactic step count, but initializing all the coefficients, etc., would be a struggle in itself.
Is this the best that can be done, or are there less-usual methods that would win out at extremely high accuracy levels?