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Suppose on a 2d lattice we have discretized some PDE and get something like $$ U_{ij}^{n+1}=\mu(u_{i-1j}^n +u_{i j-1}^n+u_{i+1j}^n+u_{ij+1}^n)+(1-4\mu)u_{ij}^n $$

What then might be meant with a statement like "$U_{ij}$ satisfies a maximum principle?"

And why might it be important that it satisfies a maximal principle?

Of course you can only guess but maybe this already helps me.

mathfemi
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  • Presumably this: https://en.wikipedia.org/wiki/Maximum_principle . – anomaly Jul 13 '16 at 15:10
  • Was my first thought, too. But sometimes "Maximum principle" also means something like that the solution is bounded from below and above? – mathfemi Jul 13 '16 at 15:15
  • In the context of PDEs, it's probably an analogue of the maximum principle for harmonic functions. – anomaly Jul 13 '16 at 15:20
  • One reason for the importance of such a property that immediately comes into my mind is the easy identifiability of the malfunction/incorrect implementation of the numerical scheme. For example, given some boundary condition, if the numerical solution anywhere exceeds this then there is clearly something wrong with the numerical scheme. – okrzysik Jul 13 '16 at 15:22

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