Monotonicity of control volume methods |
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Authors: | J M Nordbotten I Aavatsmark G T Eigestad |
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Institution: | (1) Department of Mathematics, University of Bergen, Bergen, Norway;(2) Centre for Integrated Petroleum Research, University of Bergen, Bergen, Norway |
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Abstract: | Robustness of numerical methods for multiphase flow problems in porous media is important for development of methods to be
used in a wide range of applications. Here, we discuss monotonicity for a simplified problem of single-phase flow, but where
the simulation grids and media are allowed to be general, posing challenges to control-volume methods. We discuss discrete
formulations of the maximum principle and derive sufficient criteria for discrete monotonicity for arbitrary nine-point control-volume
discretizations for conforming quadrilateral grids in 2D. These criteria are less restrictive than the M-matrix property.
It is shown that it is impossible to construct nine-point methods which unconditionally satisfy the monotonicity criteria
when the discretization satisfies local conservation and exact reproduction of linear potential fields. Numerical examples
are presented which show the validity of the criteria for monotonicity. Further, the impact of nonmonotonicity is studied.
Different behavior for different discretization methods is illuminated, and simple ideas are presented for improvement in
terms of monotonicity. |
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Keywords: | 65N12 65N06 76S05 35R05 |
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