[515.3.1] The coupled system of nonlinear partial differential and algebraic equations is solved numerically using an adaptive moving grid solver [44, 7, 12]. [515.3.2] Its FORTRAN implementation was described in . [515.3.3] Space is discretized by finite differences. [515.3.4] The time integration within this algorithm is performed using the public domain differential algebraic solver DASSL, which is a higher order implicit backward Euler scheme.
[515.4.1] The mathematical model does not fit exactly the structure of the solver. [515.4.2] The following reformulations and adaptations had to be imposed: Elimination of time derivatives, regularizations of saturation, flux symmetrization, and pressure stabilization using the pressure projection method . [515.4.3] In addition an adaptive algorithm contains parameters regulating the spatial smoothing or the adaptivity of the moving grid. [515.4.4] Details of the numerical implementation will be given elsewhere . [515.4.5] Although a thorough sensitivity analysis with respect to all artifical parameters shows that the solutions are insensitive to the parameters over a wide range, it is clear that the numerical solutions presented below are approximate at best. [515.4.6] They are given mainly to illustrate qualitatively the differences between the present theory and the accepted traditional theory.