An Aggregation-based Nonlinear Multigrid Solver for Two-phase Flow and Transport in Porous Media

TL;DR

This paper introduces a scalable nonlinear multigrid solver for two-phase flow in porous media, leveraging aggregation-based coarse spaces within a full approximation scheme to improve robustness and efficiency over traditional methods.

Contribution

The paper develops a novel aggregation-based multigrid method for two-phase flow that maintains scalability and robustness, especially for highly nonlinear problems and high CFL numbers.

Findings

Outperforms Newton's method in high CFL scenarios

Robust for highly nonlinear two-phase flow problems

Applicable to unstructured grids

Abstract

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.