Two-Dimensional Fourier Continuation and Applications

This paper presents a fast “two-dimensional Fourier continuation” (2D-FC) method for construction of biperiodic extensions smooth nonperiodic functions defined over general two-dimensional domains. The approach, which runs at cost (\mathcalO(N?og N)\) operations an $N$-point discretization grid, can be directly generalized to domains any given dimensionality, but such generalizations are not considered in this contribution. 2D-FC produced two-step procedure. In the first step one-dimensional continuation is applied along discrete set outward boundary-normal directions produce, directions, continuations that vanish outside narrow interval beyond boundary. Thus, algorithm produces “blending-to-zero normals” function values. second step, extended values evaluated on underlying Cartesian grid by means efficient, high-order interpolation scheme. A expansion then obtained direct application transform (FFT). Algorithms arbitrarily high order accuracy method. usefulness and performance proposed illustrated with applications Poisson equation time-domain wave within bounded domain. As part these examples novel “Fourier forwarding” solver introduced which, propagating plane waves as they would free space relying certain boundary corrections, solve other hyperbolic partial differential equations computing costs grow sublinearly size spatial discretization.