Computational Mechanics

Finite-Volumes Method (FVM) was first introduced independantly by Mc Donald (1971) and Mc Cormack and Paullay (1972).

This method applies to conservation laws and is based on the discretization in the physical space of the conservation laws written in an integral form. The initial continuous form is not the local partial derivative equation but the integral conservative form (integral balance of the local form over a domain). FVM are distinguished by the choice of the control volume, the localisation of the variable and the evaluation of the fluxes on the control surface (boundaries of the each control volume).

The finite-volume method is mainly used to solve convection-diffusion equations in the following forms:[ref-1]

$u_t+f_x=0$

Within each control unit ($\left[t^n, t^{n+1}\right] \times\left[x_{i-1 / 2}, x_{i+1 / 2}\right]$), it has to satisfy the integral solution, that is, integrate $t$, $x$:

$\begin{aligned} & \int_{t^n}^{t^{n+1}} \int_{x_{i-1 / 2}}^{x_{i+1 / 2}}\left(u_t+f_x\right) d t d x \\ = & \int_{x_{i-1 / 2}}^{x_{i+1 / 2}}\left(\int_{t^n}^{t^{n+1}} u_t d t\right) d x+\int_{t^n}^{t^{n+1}}\left(\int_{x_{i-1 / 2}}^{x_{i+1 / 2}} f_x d x\right) d t \\ = & \int_{x_{i-1 / 2}}^{x_{i+1 / 2}}\left(u^{n+1}-u^n\right) d x+\int_{t^n}^{t^{n+1}}\left(f_{i+1 / 2}-f_{i-1 / 2}\right) d t \\ = & 0 \end{aligned}$

Denote $\bar{u}=\frac{1}{\Delta x} \int_{x_{i-1 / 2}}^{x_{1+1 / 2}} u d x$, divide both sides of the equation by $\Delta t \Delta x$, we can deduce:

$\frac{\bar{u}^{n+1}-\bar{u}^n}{\Delta t}+\frac{1}{\Delta x} \int_{t^n}^{t^n+1}\left(f_{i+1 / 2}-f_{i-1 / 2}\right) d t=0$

In the discrete form, it still holds that the change of $u$ within the control unit over time $\Delta t$ is equal to the sum of the outflow flux $f_{i+1 / 2}$ on the right side and the inflow flux $f_{i-1 / 2}$ on the left side. That's what finite-volume method is all about.

Different results can be obtained by using different integral forms. For the simplest example, use the midpoint instead $\bar{u}=\frac{1}{\Delta x} \int_{x_{i-1 / 2}}^{x_{1+1 / 2}} u d x=u_i$, then we have:

$\frac{u_i^{n+1}-u_i^n}{\Delta t}+\frac{1}{\Delta x} \int_{t^n}^{t^n+1}\left(f_{i+1 / 2}-f_{i-1 / 2}\right) d t=0$

When the accuracy of the numerical methods for partial differential equation is less than the second order, the finite-volume method method and the finite difference method behave so similarly in some respects that in some applications they may produce very similar results, thus in this situation, do not emphasize the difference between the two.

Specifically, the finite-volume method is based on the conservation of the equations, the core of which is the integration of the conservation laws on the control volumes, while the finite-difference method is derived by approximating the derivatives of the equations. Although they have different starting points, at second-order or lower precision, they may lead to similar difference equations, so their solutions will be very similar.