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We design, analyse and implement an arbitrary order scheme applicable togeneric meshes for a coupled elliptic-parabolic PDE system describing miscibledisplacement in porous media. The discretisation is based on severaladaptations of the Hybrid-High-Order (HHO) method due to Di Pietro et al.[Computational Methods in Applied Mathematics, 14(4), (2014)]. The equationgoverning the pressure is discretised using an adaptation of the HHO method forvariable diffusion, while the discrete concentration equation is based on theHHO method for advection-diffusion-reaction problems combined with numericallystable flux reconstructions for the advective velocity that we have derivedusing the results of Cockburn et al. [ESAIM: Mathematical Modelling andNumerical Analysis, 50(3), (2016)]. We perform some rigorous analysis of themethod to demonstrate its $L^2$ stability under the irregular data oftenpresented by reservoir engineering problems and present several numerical teststo demonstrate the quality of the results that are produced by the proposedscheme.

An arbitrary order scheme on generic meshes for miscible displacements in porous media