Elliptic,Parabolic,Hyperbolic,FVM,FDM,FEM
 

Hello dear members,
I just registered here and this is my very first post to find the answer of a simple but important question.
I know this issue has been much discussed over here and any other similar websites and I read many pages, but some one please explain the answer of my question very simply and informative, I am really puzzled.
we have three types of pde , regarding to the type it may have 2,1 or zero real answer.
I can recognize the type of pde by some simple operation.
the question is that: how do you make a connection between this theoretical subject into cfd. I mean how do you detect if a cfd problem is parabolic, hyperbolic, elliptic. I know from the simplified N-S equations it can be detected. by I saw someone that tell about the type of problem at the first time.
--- another question: we also have different methods for discretizing the pde's, FVM,FDM,FEM. there is not any relation between type of pde and type of discretization. please help be by answering this question : what is the difference between hyperbolic/elliptic/parabolic grid generation? and what is the connection between the type of pde and type of problem to type of grid generation must be used.

on the other way, what does it mean the FV grid generation, Fe grid generation, FD grid generation.

I do nor expect anybody to answer my questions in detail, but please introduce some useful resource to help me finding my answers?

Hello, I think your questions are somehow ambigous... I try some answers:

1) elliptic, parabolic and hyperbolic mathematical character of second order PDE (first order equation is always hyperbolic) are referred to the existence in real space of the characteristic lines. This is referred as to continuous equations, not discrete ones.

2) CFD is devoted to "translate" the continuous PDE in a set of discrete equations that can be solved using a computer.

3) the mathematical character of the original PDE equation drives CFD user to a correct definition of the boundary domain and boundary condition. Furthermore, can address the possibility to have non regular solutions for hyperbolic equations.

4) FD, FV, FEM, SM are methods to discretize the continuos equations, you can generally use one of them depending on your experience. However, FD and SM discretize the differential form of the equations, FV and FEM are based on the weak form of the equations

Some additional answers:

1) Unless something bad is done in discretization/modeling, the character of the discretized PDE should be the same of the original PDE

2) Hyperbolic, Parabolic and elliptic grid generations are so termed because, very roughly speaking, grid points in space are created by solving, in the computational space, an hyperbolic, Parabolic or elliptic toy problem for the node coordinates of your grid. I don't know of any specific relation between the grid generation method (which can be also different, e.g., for an unstructured grid).

3) The different methods (FV, FD, FE, SM) do not, generally, require a specific grid generator (except that some FD and SM may have some serious constraints on the allowed grids). They actually differ in the way they allocate the variables on the given grid. Some specific methods (High order FE/Spectral Element Methods), may require additional points along the edges of the grid connecting the single nodes, which may be generally curve.

Paolo, concerning #1 the issue is quite complex even if you do a right discretization... Think about the simple case of the linear wave equation (first order hyperbolic) discretized by a consistent and stable FTUS scheme. The consequent modified equation is no longer first order and becomes parabolic for any non-vanishing value of dt,dx,dy,dz ...