A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

In this paper, we construct several high-order and physics-preserving numerical schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media based on the modified generalized scalar auxiliary variable (mGSAV) approach with new relaxation and the Lagrange multiplier (LM) method with the well-known Karush-Kuhn-Tucker (KKT) conditions. We construct high-order implicit-explicit BDF-k schemes with first to fifth orders for the system with homogeneous injection/production rate and boundary condition. Due to the fact that high-order BDF-k schemes with $k=2,3,4,5$ are difficult to preserve mass conservation for both phases for the system with inhomogeneous injection/production rate and boundary condition, the first- and second-order schemes are proposed based on the backward Euler and Crank-Nicolson discretizations for the mass conservation constraint equation. The constructed schemes only need to solve one linear system and a nonlinear algebraic equation with negligible computational cost at each time step. We also prove that the proposed schemes are energy stable, mass-conservative and bounds-preserving for each phase without any restrictions of time step size. Finally, various interesting numerical examples are presented to verify the accuracy and efficiency of the proposed schemes.