  In this paper, we propose a method, that is based on equivariant moving
frames, for development of high order accurate invariant compact finite
difference schemes that preserve Lie symmetries of underlying partial
differential equations. In this method, variable transformations that are
obtained from the extended symmetry groups of PDEs are used to transform
independent and dependent variables, and derivative terms of compact finite
difference schemes (constructed for these PDEs) such that the resulting schemes
are invariant under the chosen symmetry groups. The unknown symmetry parameters
that arise from the application of these transformations are determined through
selection of convenient moving frames. In some cases, owing to selection of
convenient moving frames, numerical representation of invariant (or symmetry
preserving) compact numerical schemes, that are developed through the proposed
method, are found to be notably simpler than that of standard, noninvariant
compact numerical schemes. Further, the order of accuracy of these invariant
compact schemes can be arbitrarily set to a desired order by considering
suitable compact finite difference algorithms. Application of the proposed
method is demonstrated through construction of invariant compact finite
difference schemes for some common linear and nonlinear PDEs (including the
linear advection-diffusion equation in 1D/2D, the inviscid and viscous Burgers'
equations in 1D). Results obtained from our numerical simulations indicate that
invariant compact finite difference schemes not only inherit selected symmetry
properties of underlying PDEs, but are also comparably more accurate than the
standard, non-invariant base numerical schemes considered here.
