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]]>A couple questions/notes: The title includes “implicit Euler-Method”, but this seems to be explicit Euler. Is the intention to use an implicit time stepper eventually, or is this just a typo in the title?

You mention parallelization as the next step for speed-up, but as you noted the runtime with this approach is O(N^2). Wouldn’t it be better to work on improving the scalability first with something like Barnes-Hut or a Fast Multipole Method for O(N log N) or O(N) scaling? Parallelization doesn’t really solve the underlying problem.

A final note/question: Why is your measured performance so poor? I thought your benchmarked times seemed odd so I threw together a simple Matlab script:

Runtime for 12.8k particles was 1.74 seconds, 25.6k particles was 6.3 seconds; this is nearly 7 times faster than your benchmarks. Perhaps it is worth looking at using a Linear Algebra package instead of the STL for fast vector/matrix operations?

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]]>It would change the second equation so it matches the first one. Just a minor nitpick đź™‚

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