10.4122/1.1000000554
Manzini, Gianmarco
Gianmarco
Manzini
marco.manzini@imati.cnr.it
Putti, Mario
Mario
Putti
putti@dmsa.unipd.it
Manzini, Gianmarco
Gianmarco
Manzini
marco.manzini@imati.cnr.it
Finite Volume Solutions of Strongly Anisotropic Porous Media Flow
XVI International Conference on Computational Methods in Water Resources
2006
2006
Anisotropic (i.e. direction dependent) porous media flow equations are characterized
by conductivities that may be space dependend full rank tensors. This class of
problems is also known as parameter dependent problems, the parameter being the
anisotropy ratio, i.e. the ratio between the smallest and largest eigenvalues of the
conductivity tensor. Efficient numerical discretization of strongly anisotropic
problems is generally obtained by means of ad hoc, mesh dependent scheme
modifications developed to overcome the problem known as parametric locking.
Locking is experimentally observed when the discretization error does not decrease at
the expected rate for limiting values of the parameter. This loss of convergence
disappears for sufficiently fine discretizations, but may involve costly or even
unfeasibly large calculations. The numerical solution of this type of problems
requires careful consideration of the errors that may be introduced by the
discretization scheme.
We propose a modification of Finite Volume approach based on the definition of a
diamond cell that makes optimal use of the reconstruction algorithm to yield an
accurate discretization of tangential gradients, a key ingredient for achieving
robustness of the numerical scheme for large values of the anisotropy ratio.
Numerical results are presented to show the performance of the proposed scheme.
Without introducing additional nonconsistent terms in the numerical scheme, as is
typically done in these cases (the so called variational crimes), the proposed
scheme robust with respect to the anisotropy ratio, as per the definition of Babuska
et al, 1992. We also show that the region of convergence of the method is defined by
a quadratic relationships between the anisotropy ratio and the mesh size parameter.