Elliptic,Parabolic,Hyperbolic,FVM,FDM,FEM

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 October 20, 2013, 05:48 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 1,596 Rep Power: 22 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

 October 21, 2013, 06:51 #3 Senior Member     Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 531 Blog Entries: 14 Rep Power: 17 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.

October 21, 2013, 07:02
#4
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,596
Rep Power: 22
Quote:
 Originally Posted by sbaffini 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 ...

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