  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.
