Predictive numerical simulations of subsurface processes require not only more sophisticated physical models but also more accurate and reliable discretization methods for these models. We study a new monotone finite volume method for diffusion problems with a heterogeneous anisotropic material tensor. Examples of anisotropic diffusion includes diffusion in geological formations, head conduction in structured materials and crystals, image processing, biological systems, and plasma physics. Development of a new discretization scheme should be based on the requirements motivated by both practical implementation and physical background. This scheme must
The discretization methods used in existing simulations, such as Mixed Finite Element (MFE) method, Finite Volume (FV) method, Mimetic Finite Difference (MFD) method, Multi Point Flux Approximation (MPFA) method, satisfy most of these requirements except the monotonicity. They fail to preserve positivity of a continuum solution when the diffusion tensor is heterogeneous and anisotropic or the computational mesh is strongly perturbed. Monotonicity is a very important as well as a difficult requirement to satisfy.
The mixed form of the diffusion equation includes the mass conservation equation and the constitutive equation:
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All the methods mentioned above use the same discretization of the mass
conservation equation
and differ by their approximation of the flux (constitutive) equation.
In the nonlinear finite volume scheme a reference point is defined for
each mesh cell
to approximate the concentration
. The position of
the reference point
depends on the geometry of
and value of the diffusion tensor.
For isotropic diffusion tensors and triangular cell
, the center of
the inscribed circle
is the acceptable position for the reference point.
The flux is approximated at the middle of each mesh
edge using a weighted difference of concentrations in two neighboring
cells. Nonlinearity comes from the fact that these weights
depend on a solution. Therefore the nonlinear finite volume method
results in a nonlinear algebraic system.
This system is very sparse and the dimension is equal to the number of mesh cells
.
For triangular meshes, the matrix of this system has at most four
non-zero elements in each row.
To solve the nonlinear algebraic problem we use the Picard iterative
method which guarantees monotonicity of the
discrete solution for all iterative steps. The convergence of nonlinear
iterations is a challenge problem in the case of highly anisotropic
diffusion.
The computational results demonstrate the flexibility and accuracy of the scheme.
For sufficiently smooth solutions, we achieve the
second-order convergence for concentration and at least the first-order for flux
in a mesh-dependent
-norm. For non-smooth, highly anisotropic solutions
we observe at least the first-order convergence for both unknowns.
The solution remains non-negative on different types of meshes and for different
directions of anisotropy.
Both
and
methods produce negative values.