In the translation of physical space to computational space of finite volume method,we knew that the Jacobian determinant is equal to the volume of mesh,but why? and why the area vector of a mesh is equal to the Jacobian determinant divide then gradation of computational direction variable?? who can tell me in details ,better in mathematial formulas

In the Finite Volume method, the computational and physical spaces are the same, either you are confused or misinformed, Read the book by Blazek

No, the integral form of the governing equations can be transformed just like the differential form. So, there can be a difference between computational and physical spaces in a finite volume method.

no idea what u r talking of... as far as I know there is no transformation involved in finite VOLUME method... r u speaking of finite ELEMENT/DIFFERENCE method?

The difference between computational and physical spaces comes about when you transform the governing equations from physical space (x,y,z,t) to computational space (uniform i,j,k,tau). Finite volume methods come about from discretization of the integral form of the governing equations. Finite difference methods come about from the discretization of the differential form of the governing equations. Either form of the equations can be transformed to computational space. Hence, thre can be a difference between computational and physical space for finite volume methods. There is a clear advantage to using the transformed equations for finite difference methods since the difference expressions are much simpler. The advantages of using transformed equations for finite volume methods are more subtle, but there are some (I forget what they are off the top of my head).

You may have to search the literature if this explanation doesn't clear it up for you.

>no idea what u r talking of... as far as I know there is no transformation involved in finite VOLUME method

i am not agree.

Transformation of governing equation is independent from the numerical solution method,

Dealing with curvilinear structured grids, it is formal to map the physical domain into a computational domain with uniform Cartesian (or cylindrical or spherical) grid, so the Jacobean and transformation metrics (consider christoffel symbols) are appeared in equations. Then any solution method can be used to solve mapped equations, either FVM or FDM !

a good ref. is: ZUA Warsi, Fluid Dynamics: Theoretical and Computational Approaches

yes,i am just dealing with curvilinear structured grids in FVM. THANKS FOR YOUR HELP.BUT A CAN NOT GET THE REFERENCE: Theoretical and Computational Approaches .WHAT A PITY

hi!

for FDM the geometry itself transorms and so the gov. eqns too.we need to find out the functions which represents these trasormations.

for FEM we do the co-ordinate transormations but not for the geometry, but only for evaluating the co-efficient matrices in the stiffness matrix.(element interpolation fuctions will be derived in the local co-ordinate system, and so we need to map them into global co-ordinates).

for FVM it is for finding out the area and volumes of the elements which are actually represented in the local co-ordinate systems.obviously gaussian quadrature will be used to find them.

thanks....raj.

note that after mapping there is not essential difference between fvm treatment of mapped equation with famous fvm on traditional form of eq., i was not deep on this subject, but i read some material about this that only mention them:

in C.P. Hong book detals of solution of NS equation on mapped equation is presented:

Computer Modelling of Heat and Fluid Flow in Materials Processing, Hong, C.P., 2004, CRC Press

if you don't access to this book you can read his papers that are published in : ISIJ International journal (is freely online available, search with c. p. hong keyword)

note that this is not the basic ref. but is what that i know (his book is introductury and good for begineer)

thank you for your help