The numerical simulation of subsurface processes requires efficient and robust methods due to the large scales and the complex geometries involved. To resolve such complex geometries, corner-point grids are the industry standard to spatially discretize geological formations. Such grids include non-planar, non-matching and degenerated faces. The standard scheme used in industrial codes is the cell-centered finite-volume scheme with two-point flux (TPFA) approximation, an efficient scheme that produces unconditionally monotone solutions. However, large errors in face fluxes are introduced on unstructured grids. The authors present a nonlinear finite-volume scheme applicable to corner-point grids, which maintains the monotonicity property, but has superior qualities with respect to face-flux accuracy. The scheme is compared to linear ones for complex flow simulations in realistic geological formations [1,2]. In addition, we present recent developments regarding convergence analysis for a family of nonlinear finite-volume schemes .
 Schneider, M., Flemisch, B., & Helmig, R. (2017). Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media. International Journal for Numerical Methods in Fluids, 84(6), 352-381.
 Schneider, M., Gläser, D., Flemisch, B., & Helmig, R. (2017). Nonlinear Finite-Volume Scheme for Complex Flow Processes on Corner-Point Grids. In International Conference on Finite Volumes for Complex Applications (pp. 417-425). Springer, Cham.
 Schneider, M., Agélas, L., Enchéry, G., & Flemisch, B. (2017). Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. Journal of Computational Physics, 351, 80-107.
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