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Context

I ask this question because I'm currently working on a program that solves the Cahn-Hilliard equation in 2D. For this project I need a subroutine to calculate the free energy functional by numerical integration. While looking at different options to evaluate the pros and the cons I found an library that has two possibilities: Romberg Integration and Newton-Cotes of Second Order (hereafter referred to as RI and NC2).

Questions

After reading that section I left with several doubts:

  • Romberg integration seems like a more convoluted method: It applies NC1 and then Richarson extrapolation. Does this mean it's more accurate?
  • Is there a way to determine how exact is an approximation scheme is? For example, if $\Delta x$ is the lattice spacing, can you determine if the error is of order $(\Delta x)^n$ or something similar?

The last one is a little bit out of scope of the question title but I didn't add it because the title would've been way too long:

  • Are higher order approximations better for a general smooth function (obviously for a monomial it would be it's own order)?

Thank you very much.

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  • $\begingroup$ Higher order approximations in the absence of dynamical effects are usually better up to a certain point where floating point errors dominate over truncation errors. In the presence of dynamical effects the situation is dramatically more complicated. In any case, the better thing to do is usually not to use a uniform mesh, but rather to use some kind of adaptive mesh refinement method. This is because the "difficulty" of integrating a given sub-region of the integration domain usually varies quite significantly over the domain. Romberg integration lends itself very nicely to this approach... $\endgroup$ – Ian Sep 28 '18 at 18:55
  • $\begingroup$ (Cont.) because it essentially hands you an error estimate built in. $\endgroup$ – Ian Sep 28 '18 at 18:58

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