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
Ci at time t, which we denote with
Qit, 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 Fi-1/2t approximates the average flux
along the interface xi-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 xi-1/2 for example,
depends on q(xi-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 qt + uqx = 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
xi-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,
qx) = -beta(x) qx). 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 qt + 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.
Release Notes:
.PDE.UsersGuide.FiniteVolumeMethod.Introduction