acerfon
commented
5 years ago Dear all,
I do not think symplectic integration would lead to faster field line tracing. For a given accuracy, a non-symplectic ODE solver will be faster than a symplectic one - I believe it is almost a theorem.
If you are guaranteed to have flux surfaces, such as in tokamaks, then symplectic integration makes a lot of sense, because you can afford to lose a lot of accuracy while still being sure to remain on the flux surface thanks to the symplectic nature of the solver. However, for general stellarator magnetic fields, it is relatively easy to be much more accurate with a non-symplectic solver than with a symplectic one.
Have you thought of spectral deferred correction (SDC) schemes? I believe they are close to being the state-of-the-art ODE solvers for these tasks; this is what we use for field line tracing at Courant.
Regards, Antoine
On Thu, Oct 10, 2019 at 3:33 AM Christopher Albert notifications@github.com wrote:
After discussion at ISHW with @jonathanschilling https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_jonathanschilling&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=meEavcbFdc0ufv1PWzptlA&m=U4XSQqsKcZFfE4ZqiBIVAJrxW3jlnGk5aCNFm1lk8_I&s=V5pegu5wevBx7AfAqQ75y9MfFwFL5np7hMYslR_Mcq8&e= we had the idea to speed up field line tracing by using symplectic integration.
Another, possibly even faster and simpler alternative would be to construct an interpolation of the single-turn map between Poincaré cuts and then evaluate the map for the number of turns. Construction requires 1e4-1e6 field lines to be traced over a single toroidal turn with the existing Runge-Kutta scheme and can be done in parallel.
See e.g. Kasilov et al, Phys. Plasmas 9, 3508 (2002); https://doi.org/10.1063/1.1493793 https://urldefense.proofpoint.com/v2/url?u=https-3A__doi.org_10.1063_1.1493793&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=meEavcbFdc0ufv1PWzptlA&m=U4XSQqsKcZFfE4ZqiBIVAJrxW3jlnGk5aCNFm1lk8_I&s=4w_3Lr0du9IWvgmdLxvEL19hI6RgU3qt8wZvR7eOhus&e= which is already much more complicated as particle orbits are mapped instead of field lines. This is also related to #47 https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_PrincetonUniversity_SPEC_issues_47&d=DwMFaQ&c=slrrB7dE8n7gBJbeO0g-IQ&r=meEavcbFdc0ufv1PWzptlA&m=U4XSQqsKcZFfE4ZqiBIVAJrxW3jlnGk5aCNFm1lk8_I&s=cp4bl3TuENHq9pAI6RfOIDAcJ8QdrbKgNW1j8N5xc5c&e= as on can start tracing at arbitrary theta based on such a map.
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krystophny
jloizu
After discussion at ISHW with @jonathanschilling we had the idea to speed up field line tracing by using symplectic integration.
Another, possibly even faster and simpler alternative would be to construct an interpolation of the single-turn map between Poincaré cuts and then evaluate the map for the number of turns. Construction requires 1e4-1e6 field lines to be traced over a single toroidal turn with the existing Runge-Kutta scheme and can be done in parallel.
See e.g. Kasilov et al, Phys. Plasmas 9, 3508 (2002); https://doi.org/10.1063/1.1493793 which is already much more complicated as particle orbits are mapped instead of field lines. This is also related to #47 as on can start tracing at arbitrary theta based on such a map.