A quasi-positive family of continuous Darcy-flux finite-volume schemes with full pressure support. (English) Zbl 1231.76178
Summary: A new family of flux-continuous, locally conservative, finite-volume schemes is presented for solving the general tensor pressure equation of subsurface flow in porous media. The new schemes have full pressure continuity imposed across control-volume faces. Previous families of flux-continuous schemes are point-wise continuous in pressure and flux.
When applying the earlier point-wise flux-continuous schemes to strongly anisotropic full-tensor fields their failure to satisfy a maximum principle (as with other FEM and finite-volume methods) can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution.
An \(M\)-matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9-point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field.
The full pressure continuity schemes are shown to possess a larger range of flux-continuous schemes, than the previous point-wise counter parts. For strongly anisotropic full-tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points (previously out of range) that are shown to minimize spurious oscillations in discrete pressure solutions. The new formulation leads to a more robust quasi-positive family of flux-continuous schemes applicable to general discontinuous full-tensor fields.
When applying the earlier point-wise flux-continuous schemes to strongly anisotropic full-tensor fields their failure to satisfy a maximum principle (as with other FEM and finite-volume methods) can result in loss of local stability for high anisotropy ratios which can cause strong spurious oscillations in the numerical pressure solution.
An \(M\)-matrix analysis reveals the upper limits for guaranteeing a maximum principle for general 9-point schemes and aids in the design of schemes that minimize the occurrence of spurious oscillations in the discrete pressure field.
The full pressure continuity schemes are shown to possess a larger range of flux-continuous schemes, than the previous point-wise counter parts. For strongly anisotropic full-tensor cases it is shown that the full quadrature range possessed by the new schemes permits these schemes to exploit quadrature points (previously out of range) that are shown to minimize spurious oscillations in discrete pressure solutions. The new formulation leads to a more robust quasi-positive family of flux-continuous schemes applicable to general discontinuous full-tensor fields.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
Keywords:
family of flux-continuous schemes; locally conservative; finite-volume; control-volume distributed (CVD); multi-point flux approximation (MPFA); positivity; monotonicity; maximum principle; \(M\)-matrix; full pressure continuity; anisotropy and pressure equationReferences:
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