This chapter explains how one can formulate nonlinear finite-volume (NFV) methods, as advanced discretization schemes, to solve the flow equation in porous media. These schemes are of particular interest because apart from being consistent, they are monotone by design. We explain the basic ideas of the NFV methods: how to construct one-sided fluxes, interpolate using harmonic averaging points, and obtain unique discrete fluxes through grid faces with convex combinations of one-sided fluxes. We outline key functions in the accompanied nfvm module in the MATLAB Reservoir Simulation Toolbox (MRST) and show some examples of how the method is applied.

We show that the concept of entropy is inescapable even in “pure” branches of mathematics such as algebra, topology and category theory. Specifically, we describe an entirely general categorical construction which, when given the real line and the standard simplices as inputs, produces Shannon entropy as the output. The construction involves operads and their algebras. We then show how this categorical line of thinking leads naturally to the entirely concrete and explicit characterization of information loss in Chapter 10.