Environmental Engineering Reference
In-Depth Information
Chapter 21
Numerical Methods: Finite Differences
It was demonstrated in previous chapters that the combination of fundamental
physical laws, as the conservation of mass or conservation of energy in combination
with empirical laws, such as Darcy's Law, Fick's Law or Fourier's Law, leads to
a mathematical equalities expressed as partial differential equations (pdes) - or the
ordinary differential equations, which can be considered here simple as a specific
simpler type of pde. With the addition of initial and/or boundary conditions a partial
differential equation usually has a specific solution (which not be always unique,
but this should not be a question here).
In the described way many application cases are transformed to the problem of
finding the solution of a differential equation. So far we have given several
examples of such solutions. In Chap. 1 a very simple differential equation was
solved by programming the analytical solution, describing exponential growth.
In Chap. 4 we gave the analytical solution for the 1D transport equation.
In the first edition of the topic we presented all types of solutions, as just listed,
which are given by explicit formulae of mathematical analysis. This type of
solutions is gathered under the term
analytical solutions
. However, analytical
solutions exist only for a limited set of differential equations. The solution differs
with pde and with each variation of boundary or initial conditions. They are often
valid only under special conditions concerning the coefficients of the differential
equations. If the coefficient varies in dependence of space, time or the function
itself, a formula derived for the constant coefficient, is not valid any more.
Another example: if the additional conditions change in space of time, a new
formula has to be sought. This may be very tedious, as the derivation of the formula
for the constant condition case is not valid for varying conditions anymore.
Modifying a simple pde with analytical solution one comes very easily to the
situation in which an analytical solution cannot be found, neither by analytical
derivation, nor by literature search. In that sense the method of analytical solutions
is rather limited.
Nevertheless, here are other solution methods, which are incomparable more
powerful and flexible. These are gathered under the term
numerical solutions
, and
there are finite differences, finite elements, finite volumes, boundary elements, just
