# Reference request: Finite difference methods on curvilinear (body fitted) grids

I was wondering if someone may be aware of some form of detailed summary (book, tutorial paper) about the use of finite difference methods on curvilinear (body fitted) grids.

I was only able to locate some lecture slides on the transformation of the PDE (http://www3.nd.edu/~gtryggva/CFD-Course/2013-Lecture-23.pdf) but what I am looking for is practical information on the application details of the theory using the finite difference method.

Thanks a lot!

## 1 Answer

Please check chapter 6 in the bible for computational fluid dynamics by Anderson.

I believe many books in computational fluid dynamics afterwards discussed this topic as well. I have the book by Ferziger and Peric, it addresses the problem on body-fitted grids as well. The program in which can be downloaded here on Springer's ftp site, codes are in Fortran 77 and use finite volume method to solve equation, which can be adapted to finite difference for it is using rectangular grids.

I read through the slides you gave briefly, IMHO it does a good job in explaining the body fitted grids, essentially what it does is changing of variables. Once you get the equation for $\xi$ and $\eta$: $$\nabla^2 f =\frac{1}{J} (q_1 f_{\xi\xi}− 2q_2 f_{\eta\xi} +q_3 f_{\eta\eta}) + (\nabla^2\xi)f_{\xi}+ (\nabla^2\eta)f_{\eta},$$ just apply the unsual finite difference discretization in $\xi-\eta$ coordinate system.

Some recent advances are mimetic finite difference method, please see here and here. It preserves the geometrical meaning of the differential operator, and deals with the unstructured grids problem in the slides you gave.