A method of immersed layers on Cartesian grids, with application to incompressible flows

•Extends the discrete Dirac delta function with an associated discrete masking function on the grid.•Creates a set of discrete identities on masked field data on the grid.•Develops the concept of an immersed layer, connecting grid differencing with jumps in fields across an immersed surface.•Derives semi-discrete partial differential equations on masked fields with immersed layer terms.•Demonstrates use of immersed layers in solution of prototype PDEs.