  The immersed boundary method (IBM) of Peskin (J. Comput. Phys., 1977), and
derived forms such as the projection method of Taira and Colonius (J. Comput.
Phys., 2007), have been useful for simulating flow physics in problems with
moving interfaces on stationary grids. However, in their interface treatment,
these methods do not distinguish one side from the other, but rather, apply the
motion constraint to both sides, and the associated interface force is an
inseparable mix of contributions from each side. In this work, we define a
discrete Heaviside function, a natural companion to the familiar discrete Dirac
delta function (DDF), to define a masked version of each field on the grid
which, to within the error of the DDF, takes the intended value of the field on
the respective sides of the interface. From this foundation we develop discrete
operators and identities that are uniformly applicable to any surface geometry.
We use these to develop extended forms of prototypical partial differential
equations, including Poisson, convection-diffusion, and incompressible
Navier-Stokes, that govern the discrete masked fields. These equations contain
the familiar forcing term of the IBM, but also additional terms that regularize
the jumps in field quantities onto the grid and enable us to individually
specify the constraints on field behavior on each side of the interface.
Drawing the connection between these terms and the layer potentials in elliptic
problems, we refer to them generically as immersed layers. We demonstrate the
application of the method to several representative problems, including
two-dimensional incompressible flows inside a rotating cylinder and external to
a rotating square.
