Construction of invariant compact finite-difference schemes

In this paper we propose a method, which 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 partial differential equations (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 is found to be notably simpler than that of standard, noninvariant compact numerical schemes. Further, the 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 one or two dimensions, the inviscid Burgers' equation in one dimension, viscous Burgers' equation in one or two dimensions, spherical Burgers' equation in one dimension, and shallow water equations in two dimensions). 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, noninvariant base numerical schemes considered here.