How to deal with curved boundary condition when using finite difference method?

Question

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.

Answer 1

Answering your last question first, do people actually use FDM for curved boundary nowadays I'd say the answer is no. In the commercial CFD world, 2nd order accurate finite volume schemes are the de-facto industry standard. One of the advantages of FV (and finite element/discontinuous galerkin approaches Jed mentioned) over FD is the much more natural handling of complex boundaries. FD does provide the foundation of a lot of numerical methods (FV included) and it's necessary to learn as a first step, but it's not advisable for large-scale complex problems.

As for dealing with complex boundaries in FD, I can think of two canonical ways, one of which is the interpolation/extrapolation method you mentioned. The other is to use body-fitted grid points in physical $(x,y)$ space with a conformal mapping to "computational" $\xi = \xi(x,y),\eta = \eta(x,y)$ space where $\Delta \xi = \Delta \eta = constant$. Then one can re-write terms like

$$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi} \frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial \eta} \frac{\partial \eta}{\partial x}$$

where the $\frac{\partial (\xi,\eta)}{\partial (x,y)}$ terms are called metric terms and can be computed at the beginning of a problem (or for a simple domain you might have an exact conformal mapping available), and the $u$ derivatives can be computed on a logically simple computational domain. This process makes the implementation of boundary conditions straightforward, but it requires generating a sufficiently smooth, nominally orthogonal curvilinear mesh.

I'd say this body-fitted grid approach is the "most popular treatment" for dealing with curved boundaries in FD, with the caveat that FD methods themselves are not very "popular" anymore for complex applications. It's rare to see them still come up in CFD literature except for on very simple domains.

Answer 2

Curved boundaries are covered in most CFD books, e.g., Chapter 11 of Wesseling or Chapter 8 of Ferziger and Peric.

While not a fundamental theoretical problem, the practical complexity of implementing boundary conditions for high-order methods on curved boundaries is a significant reason for interest in more geometrically-flexible methods such as the finite element method (including discontinuous Galerkin). Structured finite difference and finite volume grids are still used in some CFD simulations, but unstructured methods are gaining popularity and the local operations used by high-order unstructured methods are actually quite efficient, and thus may not suffer much loss in efficiency compared to similar FD methods. (Indeed, the geometric flexibility often makes them more efficient.)

Answer 3

I have worked on high precision fdm for the past n years. and I have used electrostatics -2 dim laplace's equation as the example for explicitly developing the high precision algorithms. until about 4 years ago the problems were constructed with horizontal or vertical lines points of potential discontinuity. if you google my name and fdm high precision you should find the references. but this is not your question. your question is fdm and curved boundaries. about a year ago I presented an order 8 solution in hong kong(see A Finite Difference Method for Cylindrically Symmetric Electrostatics having Curvilinear Boundaries) which created order 8 algorithms for interior points close to the boundary and these would require of course points on the other side of the boundary. the points on the other side of the boundary were put there by simply extending the mesh to the other side. having done this the question was how do you find the values of these points when relaxing the mesh. it was accomplished by integrating from the boundary (known potential) to the point using the algorithms. it was reasonably successful and reasonably accurate ~<1e-11, BUT required 103 algorithms each individually crafted and it was somewhat brittle, unstable geometries could be found. to remedy the above a solution has been found(order 8 and below) using (one!) minimal algorithm and the solution exhibits a considerable robustness. it has been submitted but would be available as a preprint by emailing me. I believe this technique would be extensible to time independent pde's(linear required) other than laplace and to dimensions higher than 2. I have not considered the time dependent problem but the technique being a power series technique should be adaptable and applicable. david