I'm trying to learn about numerically solving PDE by myself.
I've been beginning with finite difference method(FDM) for some time because I heard that FDM is the fundament of numerous numerical methods for PDE. So far I've got some basic understanding for FDM and been able to write codes for some simple PDE lay in regular region with the materials I found in the library and Internet, but what's strange is, those materials I've got usually talks little about the treatment of irregular, curved, strange boundary, like this.
What's more, I've never seen a easy way to deal with the curved boundary. For example, the book Numerical Solution of Partial Differential Equations - An Introduction (Morton K., Mayers D), which contains the most detailed discussion (mainly in 3.4 from p71 and 6.4 from p199) I've seen until now, has turned to a extrapolation that is really cumbersome and frustrating for me.
So, as the title asked, as to the curved boundary, usually how do people deal with it when using FDM? In other words, what's the most popular treatment for it? Or it depends on the type of PDE?
Is there a (at least relatively) elegant and high-precision way to deal with the curved boundary? Or it's just an inevitable pain?
I even want to ask, do people actually use FDM for curved boundary nowadays? If not, what's the common method for it?
Any help would be appreciated.