Algorithmic Monotone Multiscale Finite Volume Methods for Porous Media Flow

TL;DR

This paper introduces a novel algorithmic framework for multiscale finite volume methods that guarantees monotonic, accurate, and bounded solutions in porous media flow simulations, addressing stability and sensitivity issues.

Contribution

The paper proposes an algebraic modification to multiscale finite volume methods ensuring monotonicity and boundedness without losing accuracy, and enhances the performance of MsRSB for discretized systems.

Findings

Guarantees monotone solutions across various coarsening ratios

Maintains accuracy and boundedness in multiscale simulations

Improves convergence and basis function quality in multiscale solvers

Abstract

Multiscale finite volume methods are known to produce reduced systems with multipoint stencils which, in turn, could give non-monotone and out-of-bound solutions. We propose a novel solution to the monotonicity issue of multiscale methods. The proposed algorithmic monotone (AM- MsFV/MsRSB) framework is based on an algebraic modification to the original MsFV/MsRSB coarse-scale stencil. The AM-MsFV/MsRSB guarantees monotonic and within bound solutions without compromising accuracy for various coarsening ratios; hence, it effectively addresses the challenge of multiscale methods' sensitivity to coarse grid partitioning choices. Moreover, by preserving the near null space of the original operator, the AM-MsRSB showed promising performance when integrated in iterative formulations using both the control volume and the Galerkin-type restriction operators. We also propose a new approach to enhance the performance of MsRSB for MPFA discretized systems, particularly targeting the construction of the prolongation operator. Results show the potential of our approach in terms of accuracy of the computed basis functions and the overall convergence behavior of the multiscale solver while ensuring a monotone solution at all times.