An optimization-based property-preserving limiter for two-phase flow in porous media on non-matching meshes

Abstract

Two-phase flow in porous media arises in a wide range of applications in engineering and science. A fundamental challenge in its numerical simulation is that the approximation of the saturation transport equation can violate the maximum principle, and spurious oscillations can appear even when the solution remains within physical bounds. In this work, we propose a bound-preserving anti-oscillation postprocessing limiter for discontinuous Galerkin discretizations of the two-phase flow system on non-matching meshes. An optimization-based cell average limiter enforces the prescribed bounds on the cell averages while preserving global mass by minimizing the $L^2$ distance to the original cell averages. To suppress the remaining oscillations, the cell average limiter is combined with a minmod limiter, which is activated only in the cells identified by a troubled-cell detector to avoid unnecessary loss of accuracy, and a Zhang-Shu scaling limiter subsequently enforces the bounds on point values. In addition, we prove that the cell average limiter preserves the accuracy of the numerical solution at a fixed time on non-matching meshes. Numerical experiments on homogeneous, non-homogeneous, and fractured porous media demonstrate the accuracy and robustness of the proposed method.