Embedded boundary methods are ones for which an irregular domain is represented computationally as a collection of control volumes, each of which is expressed as the intersection of rectangular Cartesian cells with the domain. Then a PDE in conservation form is discretized by integrating the PDE over control volumes, with the averages of fluxes computed using quadratures on the faces. We will give an overview of such methods, including the construction of the control volumes and associated geometric objects required for the discretizations using implicit function representations of the irregular domain; (2) the construction of stable and accurate discretizations of classical PDE, including high-order discretizations, and adaptive mesh refinement. We will illustrate the properties of these methods using a variety of applications, including shocked flows, microfluildic flows, and diffusion on a surface.
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