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Introduction

In one dimension, the finite volume method consists in subdividing the spatial domain into intervals, "finite volumes" (or cells), and approximate the integral of the function q over each
of these volumes at each time step.
Denote the i-th finite volume by

Then the approximation to the average of q in the cell C_i at time t, which we denote with Q_i^t, is

Remains the question of how to find this approximation. If we think about conservation law, we note that the average within the cell can only changes due to the fluxes at the boundaries
(if we assume that no source or sink is present in the cell). The integral form of conservation law is

If we integrate this expression in time from t to t+Delta_t, we obtain

and dividing by Delta_x we reach the form

which gives us an explicit time marching algorithm. This is more clearly seen if we rewrite the expression using the notation we introduced above:

where F_i-1/2^t approximates the average flux along the interface x_i-1/2:

As can be seen from the equation (??), in order to find the average at the next time step we need to find the fluxes at the interfaces. The flux at the interface x_i-1/2 for example,
depends on q(x_i-1/2, t), which changes with time along the interface and for which we do not know the analytical solution. For this reason we need to find some approximation to
these fluxes in order to calculate the averages at the next time step. Let us now see some simple flux approximations. Examples of fluxes:

Advection equation

Consider the advection equation q_t + uq_x = 0, where u is the fluid velocity. We have seen in the previous chapters, that the flux of
the contaminant at some point x, at some time t, could be written as uq(x, t). Consider now the flux through the interface x_i-1/2.
Inserting it into the average update rule, we obtain the finite volume method for the advection equation:

Diffusion equation

In the advection equation, the flux depends on q: f(q) = uq. The flux in the diffusion equation depends on the derivative of q:

where beta is the conductivity. If beta is space dependent then the flux will depend on space too (f(x, q_x) = -beta(x) q_x). In the following we will assume for simplicity
that beta is constant. Now remains the question of how to approximate numerically the diffusion flux. One possibility were:

By inserting this flux approximation into the average update rule (??), we obtain:

It is interesting to note, that after some algebraic manipulations, we can write the average update rule in the form

which is equivalent to the finite difference discretization of the conservation law equation q_t + f(q)_x = 0. As said in (LeVeque): Many methods can be equally well viewed as
finite difference approximations to this equation or as finite volume methods. Another form of the average update rule is

which give us an ODE for each average cell. This form is more suitable for the implementation in Dymola and all Finite Volume Method blocks are based on this form of update rule.

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