What are the differences between Conservative differential form, Non-conservative differential form, Conservative Integral form and Non-conservative integral form of differential equations? I know that there is a lot of different post on 'conservative vs non-conservative form' and ' differential vs integral form' but from what I have read conservative and non-conservative form of the governing equations also have a differential and integral type and none of the posts seem to address them together. So where are these different types of equations applied and what are the advantages and disadvantages of one over the other?

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    $\begingroup$ Would you mind giving some indication of your level of previous mathematical training and interest? This is a rather broad topic, and something which is vital in one field might be considered trivial in another. $\endgroup$ – origimbo Jul 1 '19 at 17:31
  • $\begingroup$ @origimbo I am a mechanical engineer and have been studying computational methods like CFD, FEM, and FVM for the past couple of months. I have understood the basics like how in conservative form the flux are inherently conserved and in nonconservative case, this may not be the case. How the integral form is like is averaging over the control volume and so it has the weak condition of continuity and the differential forms are like the strong form. I am not a mathematician. Just a curious engineer trying to understand the mathematical jargon and their inherent meanings. $\endgroup$ – GRANZER Jul 1 '19 at 18:33
  • $\begingroup$ @origimbo Can you please help me out with this? $\endgroup$ – GRANZER Jul 3 '19 at 2:38
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    $\begingroup$ It seems you have reached the point of an understanding of the technical differences already yourself. If I can find the time I'll have a go at drafting an answer giving one viewpoint of the perceived advantages, but it's likely to get long quickly. The short form would be something like " FEM, spectral and FVM are naturally integral/measure based, FDM and SPH are differential/pointwise. Conservation tends to be important for stability, discontinuities and physical relevance, nonconservative can be more easily finessed for fast convergence". $\endgroup$ – origimbo Jul 5 '19 at 12:54

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