CODEWITHSUNDEEP
numerical-methodsodeextrapolation
Paul Aner
Paul Aner

Reputation: 503

Combining the Adams-Bashforth-Method with Richardson-Extrapolation

I am experimenting with an N-Body-Simulation with the goal to be efficient with a large number of particles. That means, the most costly thing will be calls to the ODE-function. Adams-Bashforth only needs one ODE-evaluation per step, so this seems an efficient choice. We get higher orders "for free", but the method becomes unstable at higher orders.

My thought now was that I can also "for free" calculate different (bigger) step-sizes with Bashforth. Say, I run Bashforth with order 5. That means, I need 4 previous states. As long as I have stored at least 8 previous states, I can just use every other state and thus get a Bashforth-step with twice the step-size.

If I understand Richardson-Extrapolation correctly, I can combine these two results and achieve an order of 6 (or higher if I use more step-sizes) - again, "for free". Since there is no higher order Bashforth involved, there should not be any (more) instabilities, correct?

Can this approach work or am I missing something? For example, will the Bashforth-steps with higher step-size become so unstable that the Richardson-Extrapolation breaks down?

Upvotes: 0

Views: 17

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