In any numerical method for the approximation of a partial differential equation, the degrees of freedom of the method are the discrete variables that are used to approximate the solution. These may however have different interpretations. In finite difference methods and in some finite element methods, DOFs are interpreted as values of the solution at some points in space. In finite volume methods, DOFs are interpreted as approximations of the average of the solution over a control volume (in some advanced FV methods, some DOFs represent the average of the gradient of the solution). In spectral or modal finite element methods, the DOFs are the coefficients of the solution's representation in some functional space (e.g. Fourier coefficients if a Fourier basis is employed). The number of DOFs is the number of unknowns of the problem.

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/98401/fld3767.pdf?sequence=1&isAllowed=y

DOF are practically the number of nodes in the (time and space) computational grid. That is not constrained only to FV but extends to FD, too.

First: Assume PDE discretization error is defined by Taylor series expansion in your FVM discretization.

Now:

- if you have a first order method to reduce the error to half you have to refine your mesh (here mesh means both spatial and temporal descretization) by factor of two.

- if you have a second order method (in time and space) to reduce the error to half you have to refine your mesh by factor of sqrt(two).

So one

In 3D non steady methods if you want to reduce error by factor of four you have to reduce dx, dy, dz, and dt by factor of 4 (so 4x4x4x4 = 256)

So you increase the number of computational nodes by factor of 256.

Please let me know if you are not clear on that

Freedom degree is by definition all possibles states that some thing can assume in a system, e.g. in FEM a mesh node can have six freedom degree - to move in x,y,z, direction, and to rotate in the x,y,z-axi, because the displacement is a variable to be determined.

Now, consider the FVM, the nodes can have the same behavior, that is, a finite volume can has 6-fd if some displacement occurs (what is not common, but it happens). We need to consider other states. Each variable such velocities components, density, energy have a governing equation to be solved for each finite volume (in cell-center or cell vertex formulations - does not matter, it is the same thought). Hence each variable is freedom degree. e.g. incompressible flow at constant temperature: we have u,v,w, and P to be solved - four freedom degrees.

In other words, each variable represents a freedom degree, to be this variable a flow variable or a "mechanical" variable, doesn't matter.

So, we have: # finite volumes x # variables = #freedom degrees.

A detail: The # finite volumes will depends on the formulation: cell-center ( = # elements) or cell-vertex( # nodes).

Considering a incompressible flow with 3x3x3 mesh with a cell-vertex formulation we have 27 FV and and 4 variables, hence, FD = 27 x 4 = 108. With a RANS turbulence model FD=162, with energy balance FD=187, and so on...

I hope help you!

In any numerical method for the approximation of a partial differential equation, the degrees of freedom of the method are the discrete variables that are used to approximate the solution. These may however have different interpretations. In finite difference methods and in some finite element methods, DOFs are interpreted as values of the solution at some points in space. In finite volume methods, DOFs are interpreted as approximations of the average of the solution over a control volume (in some advanced FV methods, some DOFs represent the average of the gradient of the solution). In spectral or modal finite element methods, the DOFs are the coefficients of the solution's representation in some functional space (e.g. Fourier coefficients if a Fourier basis is employed). The number of DOFs is the number of unknowns of the problem.

Please go through this journal (Fig :3) ,I hope u will find solution for your problem

Dear Luca Bonaventura

Thank you for answer.

I think " different interpretations " was the most important problem which took me confused. But your explanation, helped me for better understanding.

Thank you

Dear Filippo Maria Denaro , Yogananda bharath kumar Jada , Kaveh Zamani

Thank you for answers.