Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
Predictive numerical simulations of subsurface processes require not only more sophisticated physical models but also more accurate and reliable discretization methods for these models. The discretization methods used in existing simulations fail to preserve positivity of a continuum solution of the pressure equation when the porous media is heterogeneous and anisotropic or the computational mesh is strongly perturbed to resolve complex geological structures. A negative discrete solution implies non-physical Darcy velocities and hence wrong prediction of a contaminant transport. Next, the complexity of the subsurface physics results in using of upscaled coefficients and coarse meshes which only amplify problems of existing discretization methods.
In our research we study new monotone finite volume discretization methods that guarantee positivity of the discrete solution for unstructured meshes and strongly heterogeneous anisotropic diffusion tensors. The methods are based on the nonlinear flux formula proposed by C.Le Potier in 2005. We gave the first proof of monotonicity of the new finite volume methods for a stationary diffusion problem. We also developed new monotone methods for shape-regular polygonal meshes and heterogeneous isotropic diffusion coefficients. All new methods are locally conservative, second-order accurate for smooth solutions, and have compact discretization stencils.
