35M11 Initial value problems for equations of mixed type
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We consider a Darcy flow model with saturation-pressure relation extended
with a dynamic term, namely, the time derivative of the saturation.
This model was proposed in works of J.Hulshof and J.R.King (1998), S.M.Hassanizadeh and W.G.Gray (1993),
F.Stauffer (1978).
We restrict ourself to one spatial dimension and strictly positive
initial saturation. For this case we transform the initial-boundary value
problem into combination of elliptic boundary-value problem and initial
value problem for abstract Ordinary Differential Equation. This splitting
is rather helpful both for theoretical aspects and numerical methods.