I'm working on a time integration scheme for my research. As a result, I have come across an interesting phenomenon. Somehow, the total energy of the scheme oscillates. At any given time the total energy of the system could be increasing or decreasing but does so in a oscillatory fashion. Interestingly enough, for what I'm modeling, this integration scheme works great. I'm curious if there is a way to prove whether the energy of this scheme is stable throughout the time history.

FYI, I'm a Ph.D. student in structural engineering. So, this topic is a little out of my field. I'm just curious if the scheme is stable or not, and if it can be proven. Are their other time integration schemes that don't satisfy the conservation of energy but remain stable? Thanks.

  • $\begingroup$ Like @Qmechanic, I think that this would be better at scicomp. There, you would be able to post more details about the integration scheme, and they would be more competent to help you analyze it that we would. $\endgroup$ – Colin McFaul May 14 '13 at 15:06
  • $\begingroup$ You'll have to describe your scheme, or there's little anyone can provide in terms of feedback. As for schemes that are stable but do not preserve energy: start with $\ddot x(t)=x(t)$ and define the energy as $E(t)=x(t)^2 + \dot x(t)^2$. This is the equation of a harmonic oscillator. Reformulate this as a first order system: $\dot x_1 = x_2, \dot x_2=-x_1$. If you apply the Crank-Nicolson scheme, energy is preserved. But for the implicit Euler equation, the energy decreases. Both are stable. $\endgroup$ – Wolfgang Bangerth May 16 '13 at 1:43

Integration schemes that conserve energy are a research field of their own. They are called symplectic or symplectic integrator. When you google symplectic ingegrator, you will find a lot of search hits. Unfortunately many of them are quite mathematical. Others are not mathematical at all, but may lack the necessary depth to help you with what you are doing. I think the one in this link is rather ok. There is also a Wikipedia entry on symplectic integrators that links to further references.

However, in order to actually help you, you would have to describe what you are doing and also describe the algorithm you are using. I think this question would be a better match for scientific computing stack exchange. I'll send a flag to a moderator.

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