ExpandLGF.jl provides fast and accurate evaluations of the Lattice Green's Functions (LGFs) for high-order finite difference discretizations of Poisson's equation.
Lattice Green's Functions (LGFs) are fundamental solutions to discretized linear operators that are frequently used in the numerical solution to elliptic PDEs. As such, they are a useful tool for solving Poisson's equation on domains with one or more free-space boundary conditions, which appear frequently in simulations of incompressible fluid flows (see ViscousFlow.jl for one such example).
Generally LGFs are defined through a Fourier integral with a singular and oscillatory integrand. For the standard second-order centered discretization of the Laplacian, analytical techniqes can be used to greatly simplify the 2D or 3D integral into a 1D form which is amenable to numerical quadrature. This package uses a combination of symbolic computation and numerical quadrature to extend these simplifications to higher-order discretizations.
For interactions between distant sources, a far-field asymptotic expansion of the LGF can bypass numerical integration altogether. This package implements an existing algorithm from Martinsson and Rodin to calculate these far-field expansions. The results are available either as compiled Julia functions or C code.
Finally, the LGF concept can be extended to domains with a combination of periodic and free-space boundary conditions. For domains with one unbounded dimension and one or more periodic dimensions, a C++ header that evaluates the LGFs of several common discretizations is included in the ccode folder.