[stephencanon]

The orbital example where BDF loses momentum is really about the difference between a second-order method (BDF2) and a fourth-order method (RK), rather than explicit vs implicit (but: no method with order > 2 can be A-stable; since the whole point of implict methods is to achieve stability, the higher order BDF formulas are relatively niche).

There are whole families of _symplectic_ integrators that conserve physical quantities and are much more suitable for this sort of problem than either option discussed. Even a low-order symplectic method will conserve momentum on an example like this.

[Certhas]

Obviously^1. But it illustrates the broader point of the article, even if for the concrete problem even better choices are available.

1) if you have studied these things in depth. Which many/most users of solver packages have not.

[_alternator_]

The fascinating thing is that discrete symplectic integrators typically can only conserve one of the physical quantities exactly, eg angular momentum but not energy in orbital mechanics.

[srean]

I have always wanted to know if there is any theorem that says one cannot preserve all of the standard invariants.

For example, we know for mappings that we cannot preserve angles, distances and area simultaneously.

[_alternator_]

The short answer is that discretization can generally preserve only one invariant exactly; others must be approximate.

This could provide some evidence for the universe not being truly discrete since we have multiple apparent exactly preserved kinematic quantities, but it’s hard to tell experimentally since proposed discrete space times have discretization sizes on the order of hbar, which means deviations from the continuum would be hard to detect.

[srean]

Thanks for replying on this now rather inactive thread.

I am really curious about this issue and am looking for a theorem that gives an impossibility result (or an existence result).

It might be well known but I don't know DE to be aware about f the result.

[ekelsen]

leapfrog!