Discretizing Hyperbolic Transport Equations

The chapter discusses numerical discretization of first-order quasilinear hyperbolic PDEs, so-called conservation laws. We start by briefly reviewing some of the theory for these equations, including weak solutions, discontinuities, and entropy conditions. We then present a general family of conservative finite-volume methods that includes centered as well as upwind and Godunov-type schemes. We demonstrate typical deficiencies in classical schemes including smearing of discontinuities and creation of nonphysical oscillations. We end the chapter by presenting the implicit, upstream-mobility scheme, which is the most widespread method in reservoir simulation.