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).
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.