  Stencil composition uses the idea of function composition, wherein two
stencils with arbitrary orders of derivative are composed to obtain a stencil
with a derivative order equal to sum of the orders of the composing stencils.
In this paper, we show how stencil composition can be applied to form finite
difference stencils in order to numerically solve partial differential
equations (PDEs). We present various properties of stencil composition and
investigate the relationship between the order of accuracy of the composed
stencil and that of the composing stencils. We also present comparisons between
the stability restrictions of composed higher-order PDEs to their compact
versions and numerical experiments wherein we verify the order of accuracy by
convergence tests. To demonstrate an application to PDEs, a boundary value
problem involving the two-dimensional biharmonic equation is numerically solved
using stencil composition and the order of accuracy is verified by performing a
convergence test. The method is then applied to the Cahn-Hilliard phase-field
model. In addition to sample results in 2D and 3D for this benchmark problem,
the scalability, spectral properties, and sparsity is explored.
