A Compact MPFA Method with Improved Robustness

MPFA methods were introduced to solve control- volume formulations on general grids. While these methods are general in the sense that they may be applied to any grid, their convergence properties vary. <br><br>An important property for multiphase flow is the monotonicity of the numerical elliptic operator. In a recent paper, conditions for monotonicity on quadrilateral grids have been developed. These conditions indicate that MPFA formulations which lead to smaller flux stencils, are desirable for grids with high aspect ratio or severe skewness and for media with strong anisotropy or strong heterogeneity.<br><br>We introduce a new MPFA method for quadrilateral grids termed the L-method. The methodology is valid for general media. For homogeneous media and uniform grids, this method has four-point flux stencils and seven-point cell stencils in two dimensions. The reduced stencil appears as a consequence of adapting the method to the closest<br>neighboring cells. <br><br>We have tested the convergence and monotonicity properties for this method, and compared it with the O-method. For moderate grids the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O-methods is lost, while the L-method converges with a reduced convergence rate.<br><br>The L-method has a somewhat larger monotonicity range than the O-methods, but the dominant difference is that when monotonicity is lost, the O-methods may give large oscillations, while the oscillations with the L-method are small or absent.<br>