Discretization methods preserving essential mathematical and physical properties of continuum models play critical role in multi-physics simulations. They guarantee that discrete solutions are conservative, satisfy physical bound (positivity of concentration or temperature) or maximum principles, and discrete systems preserve structure of continuum PDEs.
The latter can be leveraged by algebraic solvers. Development of structure preserving discretization methods becomes more difficult on unstructured polytopal meshes. A single discretization framework cannot satisfy all requirements imposed by physical models and a few ideas have to be combined.
In this talk, I'll present numerical schemes implemented in the DOE/EM code Amanzi simulating the reactive transport in porous media. In particular, I'll discuss the mimetic finite difference and nonlinear linear finite volume methods.