Can the numerical method for a set of ordinary differential equations be non-conservative? 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.
 A: 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.
