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Convergence order of three point central finite difference scheme

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Old   November 11, 2017, 03:26
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Filippo Maria Denaro
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Originally Posted by lmalenica View Post
My opinion is that polynomial order determines theoretical (maximum) order of approximation. Differential operator (i.e. order of PDE/ODE) should affect only accuracy of the approximation and not the order. From results what I see its seems that that formulation/method can reduce order of method but I don't know how to verify/prove it.

One more thing that confuses me here. If we consider unsteady convection equation and discretized both spatial and temporal derivatives with forward (or backward) finite difference our method is first order accurate both in space and time. Both terms are derived by using first two terms from Taylor series (linear approximation) which implies second order method when we consider solution (not derivative) convergence order. Moreover we know that Euler temporal discretization is locally (if we consider single time step) 2nd order accurate, and that loses one order because accumulation of error during time (multiple time steps). Why then is spatial discretization only 1st order and not 2nd order accurate?

1) the approximation using a p-degree polynomial must be seen on the function you are interpolating. The approximation is the same in all the interpolation interval of the p+1 nodes. When you consider derivatives the local truncation error scales differently depending on the position you evaluate the derivative. If you can understand some word in italian, that is explained in my chap.3 I attached before.

2) Your second question is not clear to me. The FTUS is first order in time and space. Again the local truncation error of the PDE shows O(dt,h). The Euler equations have pressure term, non linearity, etc. Then are you considering the one-step accuracy as illustrated in the book of Leveque?
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