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Given y'(t)=f(t,y) and y(0)=y0.

One way to parallelise the DE is to split the interval [0,a] into n subintervals [ak/n, a(k+1)/n] where k = {0,1,...,n-1}.

**Problem:**I need to know the initial conditions y(ak/n) to be able to solve the DE in parallel.

Now I can run the DE solver with a large timestep to compute y (without the precision I require) and thus extract an approximation to y(ak/n). Is there a way to use some sort of iterative scheme to refine y(ak/n) and get it as accurate as I need?

Alternatively, if you have some other method of solving ODE's in parallel, I would like to know about it too.

Thank you!