How Can Initial Conditions for Parallel ODE Solvers Be Refined for Accuracy?

[Qyzren]

I wish to solve the following DE numerically on the interval t:[0,a] using parallel processors.

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!

 

[mighty2000]

Thank you for your question. I understand the importance of solving differential equations efficiently and accurately, especially when dealing with large intervals and using parallel processors. I would like to offer some suggestions on how to approach this problem.

Firstly, you are correct in splitting the interval [0,a] into n subintervals as it allows for parallelization of the DE solver. However, as you mentioned, the initial conditions for each subinterval must be known in order to solve the DE in parallel. One way to obtain these initial conditions is to use a shooting method. This involves choosing an initial guess for y(ak/n) and then solving the DE for each subinterval using this guess. The resulting solution can then be compared to the desired boundary condition y(ak/n) and the initial guess can be adjusted accordingly. This process can be repeated until the desired accuracy is achieved. This method can be easily parallelized by assigning each subinterval to a separate processor.

Another approach is to use a multistep method, such as the Adams-Bashforth-Moulton method, which uses previous values of the solution to approximate the current value. This method can be parallelized by dividing the interval into smaller subintervals and using the previous values from neighboring subintervals to approximate the current value. However, this method may not be as accurate as the shooting method and may require more computational resources.

In terms of other methods for solving ODEs in parallel, you may also consider using a spectral method. This involves approximating the solution using a series of basis functions, such as Fourier or Chebyshev polynomials. The coefficients of these basis functions can then be solved for in parallel. This method can be very accurate, but may require more computational resources and may not be suitable for all types of differential equations.

I hope these suggestions are helpful to you. As always, it is important to carefully consider the problem at hand and choose the most appropriate method for your specific needs. Good luck with your research!