Boundary conditions

All computations take place on the unit cube \([0,1]^3\) (particle coordinates must lie inside it; for the particle FMM with periodicity, box_size rescales the period). The boundary condition is selected at tree-construction time.

The C++ interface takes a pvfmm::BoundaryType (defined in include/mpi_tree.hpp):

Value	Meaning
FreeSpace	free-space (decaying) boundary conditions
PX	periodic in \(x\); free in \(y, z\)
PXY	periodic in \(x, y\); free in \(z\)
PXYZ	periodic in all three directions
Periodic	alias for PXYZ

Periodic sums are evaluated by accumulating the multipole expansion of the periodic images into the root node’s local expansion (FMM_Pts::PeriodicBC); no Ewald-style parameter tuning is needed.

Note

For kernels without scale-invariant decay the periodic sum is defined up to the usual gauge/mean constraints; for the Laplace kernel with periodic boundary conditions the source must have zero mean over the box (as in fmm_cheb -test 1, whose source integrates to zero).

The C-level interfaces mirror the full set of boundary types through enum PVFMMBoundaryType (PVFMMBoundaryFreeSpace, PVFMMBoundaryPXYZ, PVFMMBoundaryPX, PVFMMBoundaryPXY; values 0/1 coincide with the boolean periodic flag accepted by earlier versions). The volume-tree constructors take it directly, and the particle-FMM constructor PVFMMCreateContext* takes it together with box_size (the domain length, which is the period along the periodic directions; box_size <= 0 is allowed only with free space, where the bounding box is determined from the points). The Fortran interface uses the PVFMMBoundary* constants, and the Python/Julia bindings expose an FMMBoundaryType with the same values (Python interface, Julia interface).