FDvsFV

Introduction

When it comes to discretization approaches, CFD textbooks tell the differences between Finite-Difference Method and Finite-Volume Method in their introductionary chapters. I have remembered the points below since I began learning CFD.

• The starting point of the FD method is the the differential conservation equations. The partial derivatives are approximated in term of nodal values. It’s easy to obtain higher-order schemes on regular grids.
• The FV method applied the integral form of conservation equations to a finite number of contiguous control volumes(CV). Interpolation is used to express valued at the CV surface in terms of CV center values. It’s suitable for complex gemetries.

But I haven’t fully understood these sentences until I write codes using both methods. Here I write down my understanding of both methods.

1D situation

It’s easy to illustrate 1D conservation law at the begining because it’s simple and doesn’t involve geometry and coordinate transformation.

Finite-Difference

$$
\begin{equation}
u_t + f_x=0
\end{equation}
$$

Using forward difference in time and central difference in space(FTCS):
$$
\begin{equation}
\frac{u_j^{n+1}-u_j^n}{\Delta t} + \frac{f_{j+1/2}^n - f_{j-1/2}^n}{\Delta x}=0
\end{equation}
$$

It is just Mathematics. The concepts used here are forward and central difference. Taylar expansion can be used to abtain the accuracy.

Finite-Volume

$$
\begin{equation}
\frac{\partial}{\partial t}\int u dx + \oint f dx=0
\end{equation}
$$

Let $\bar{u}=\frac{1}{\Delta x} \int_\Omega dx$ be the cell-averaged value. Let $f_{j+1/2}$ and $f_{j-1/2}$ be hte fluxes deffined at the cell boundaries.When the equation is applied to a control volume centered at $x=x_j$ , and divided by the cell volume $\Delta x$ , we obtain:
$$
\begin{equation}
\frac{d\bar{u}_j}{d t} + \frac{f_{j+1/2}^n - f_{j-1/2}^n}{\Delta x}=0
\end{equation}
$$

Using the forward Euler time integration scheme, we obtain the same fomular as the FD method.
The concepts used here are cell-averaged value and cell boundary flux. They all have physical meanings.
One more thing need to remember is that the volume-averaged value differs from the cell center value by $\mathcal{O}(\Delta x^2)$ .

3D situation

The FD method need to transform the variables in physical space $(x,y,z,t)$ to computational space $(\xi,\eta,\zeta,t)$ . In the process, metrics and jacobian of the transformation are requested.
The FV method need geometry quantities such as volume and area to compute cell-averaged value and cell boundary flux.
In this section, I will illustrate the equivalence between Mathematicial derivatives
metrics and Physical quantities areas in regular grid.

Generalized Coordinates

CFL3D manual gives a detialed three-dimensional transformation between the Cartesian variables $(x,y,z,t)$ and the generalized coordinated $(\xi,\eta,\zeta,t)$ .
Here is a simple version without moving-grid.