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.

翻译：多尺度有限体积方法因其产生的简化系统具有多点模板，可能导致非单调和超界解。我们针对多尺度方法的单调性问题提出了一种新颖解决方案。所提出的算法单调（AM-MsFV/MsRSB）框架基于对原始MsFV/MsRSB粗尺度模板的代数修正。AM-MsFV/MsRSB保证了单调且有界解，且在不同粗化比例下不牺牲精度，从而有效解决了多尺度方法对粗网格划分选择敏感性的挑战。此外，通过保留原始算子的近零空间，AM-MsRSB在结合控制体积和伽辽金型限制算子的迭代公式中展现出良好性能。我们还提出了一种新方法，针对MPFA离散系统增强MsRSB性能，特别是针对延长算子的构建。结果表明，我们的方法在计算基函数精度和多尺度求解器整体收敛行为方面具有潜力，同时始终确保单调解。