High-resolution Godunov-type methods are most commonly associated with compressible flow solutions. Their introduction to incompressible flow solutions was most directly impacted by their use with a projection method ^[1] by Bell, Colella and Glaz (BCG) [45] in 1987. BCG incorporated an exact projection with second-order Godunov methods that had then begun to reach a state of maturity with respect to compressible shock dynamic calculations. This parallels the current time where these methods are commonly used in incompressible flow calculations as well.

This chapter will describe the salient aspects of high-resolution Godunov-type methods used in conjunction with projection methods. We will begin by describing the fundamental approximations in the form of the first-order accurate time integration scheme. This will include the Riemann solutions specialized for the velocity field associated with the incompressible flow. This is followed by extensions necessary for the first-order method to produce formally second-order approximations using monotonicity limited derivatives (slopes) in one dimension. This approach is expanded upon with the discussion of genuinely multidimensional derivative approximations that are similarly limited using extensions of the one dimensional monotonicity criteria to two and three dimensions. Finally, we close with a discussion and analysis of the numerical stability of these methods for the advection-diffusion equation.