Motivated by observations of saturation overshoot this article investigatesgeneric classes of smooth travelling wave solutions of a system of two-couplednonlinear parabolic partial differential equations resulting from a flux function ofhigh symmetry. All boundary value problems of the travelling wave Ansatz, thatlead to smooth travelling wave solutions, are systematically explored. A complete, visual and computationally useful representation of the five dimensional manifold connecting wave velocities and boundary data is found by using a dynamical system approach. The travelling waves exhibit monotonic, non-monotonic or plateau-shaped behaviour. Special attention is given to the non-monotonic proles. The stability of the travelling waves is studied by numerically solving the full system of the partial dierential equations with an ecient and accurate adaptive movinggrid solver.
Hörig, O., Zegeling, P. A., Doster, F., & Hilfer, R. (2016). Non-monotonic Travelling Wave Fronts in a System of Fractional Flow Equations from Porous Media. Transport in Porous Media, 114(2), 309–340. https://doi.org/10.1007/s11242-015-0618-2