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I have a set of highly coupled ODE that I am solving. I'm just not understanding whether the quantities will be inherently conserved because it's an ODE. Everything I'm seeing on Google is for PDE's.

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    $\begingroup$ One such problem is integrating an orbit in a gravitational field. We know that the energy and angular momentum are conserved in such problems. If there are errors in the numerical integration, these constants may not be conserved. I have heard there are integration techniques that deal with this, but am not familiar with them. $\endgroup$ Dec 1, 2016 at 21:16
  • $\begingroup$ @RossMillikan : For conservative systems you can use symplectic integrators. They preserve the conserved quantities to at least the integration order, i.e., the drift away from a tightly oscillating near constant is slower than the order of the method. -- The most known and abused (in reducing it to symplectic Euler, not computing the force correctly,…) symplectic integrator is the Newton-Cromer-Störmer-Verlet method. See Hairer et al. on geometric numerical integration $\endgroup$ Dec 2, 2016 at 2:23

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Numerical methods for ODEs are more often than not found to be non-conservative when solving systems having conserved quantities. How non-conservative they are depends on things like the step-size, the order of the method and how long "time" you integrate over.

A typical example of this is found when numerically solving for plantary orbits (e.g. the Earth around the Sun). Below are some plots showing the numerical solution of such a system for different ODE solvers taken from this paper:

All three methods deviate from the true analytical solution (in blue) after enough orbits. If we look at the total energy, which should be conserved, we see that the last integrator conserves it very well while the first two don't:

Integrators that have this property is often called symplectic integrators. Such integrators can always be constructed for systems that derive from a Hamiltonian (as is often the case for physical systems).

If you are solving a system that have conserved quantities then monitoring the conservation of these is often a good way of checking the "goodness" of the numerical solution.

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