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For Example:

Modelling Processes in Porous
Media with Differential Equations

This chapter illustrates the scientific context in which differential equation
models may occur, in general, and also in a specific example. Section 0.1
reviews the fundamental equations, for some of them discretization tech-
niques will be developed and investigated in this book. In Sections 0.2 –
0.4 we focus on reaction and transport processes in porous media. These
sections are independent of the remaining parts and may be skipped by
the reader. Section 0.5, however, should be consulted because it fixes some
notation to be used later on.

0.1 The Basic Partial Differential Equation Models

Partial differential equations are equations involving some partial deriva-
tives of an unknown function u in several independent variables. Partial
differential equations which arise from the modelling of spatial (and tempo-
ral) processes in nature or technology areof particular interest. Therefore,
d
we assume that the variables of u are x =(x 1 ,...,x d ) T ∈ R for d ≥ 1,
representing a spatial point, and possibly t ∈ R, representing time. Thus
the minimal set of variables is (x 1 ,x 2 )or(x 1 ,t), otherwisewe haveordinary
differential equations. We will assume that x ∈ Ω, where Ω is a bounded
domain, e.g., a metal workpiece, or a groundwater aquifer, and t ∈ (0,T ]for
some (time horizon) T> 0. Nevertheless also processes acting in the whole
d
R × R, or in unbounded subsets of it, are of interest. One may consult the
Appendix for notations from analysis etc. used here. Often the function u