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4 0. Modelling Processes in Porous Media with Differential Equations

In more general versions of (0.7) ∂ t u is replaced by ∂ t S(u), where S
depends linearly or nonlinearly on u. In the case of heat conduction S is
the internal energy density, which is related to the temperature u via the
factors mass density and specific heat. For some materials the specific heat
depends on the temperature, then S is a nonlinear function of u.
Further aspects come into play by the source term Q if it depends linearly
or nonlinearly on u, in particular due to (chemical) reactions. Examples for
these cases will be developed in the following sections. Since equation (0.3)
and its examples describe conservation in general, it still has to be adapted
to a concrete situation to ensure a unique solution u. This is done by the
specification of an initial condition

S(u(x, 0)) = S 0 (x) for x ∈ Ω ,
and by boundary conditions. In the example of the water filled container
no mass flux will occur across its walls, therefore, the following boundary
condition
J · ν(x, t)=0 for x ∈ ∂Ω,t ∈ (0,T ) (0.9)
is appropriate, which — depending on the definition of J — prescribes the
normal derivative of u, or a linear combination of it and u. In Section 0.5
additional situations are depicted.
If a process is stationary, i.e. time-independent, then equation (0.3)
reduces to

∇· J(x)= Q(x, u(x)) for x ∈ Ω ,
which in the case of diffusion and convection is specified to
−∇ · (K∇u − cu)= Q.
For constant K —let K = 1 by scaling —, c =0, and f := Q,being
independent of u, this equation reduces to
−∆u = f in Ω ,
the Poisson equation.
Instead of the boundary condition (0.9), one can prescribe the values of
the function u at the boundary:
u(x)= g(x) for x ∈ ∂Ω .

For models , where u is a concentration or temperature, the physical reali-
sation of such a boundary condition may raise questions, but in mechanical
models, where u is to interpreted as a displacement, such a boundary con-
dition seems reasonable. The last boundary value problem will be the first
model, whose discretization will be discussed in Chapters 1 and 2.
Finally it should be noted that it is advisable to non-dimensionalise the
final model before numerical methods are applied. This means that both
the independent variables x i (and t), and the dependent one u, are replaced