FFT-Based High Order Central Difference Schemes for Poisson’s Equation with Staggered Boundaries

Abstract

This work concerns with the development of fast and high order algorithms for solving a single variable Poisson’s equation with rectangular domains and uniform meshes, but involving staggered boundaries. Here the staggered boundary means that the boundary is located midway between two adjacent grid nodes. Due to the popularity of staggered grids in scientific computing for solving multiple variables partial differential equations (PDEs), the planned development deserves further studies, but is rarely reported in the literature, because grand challenges exist for spectral methods, compact finite differences, and fast Fourier transform (FFT) algorithms in handling staggered boundaries. A systematic approach is introduced in this paper to attack various open problems in this regard, which is a natural generalization of a recently developed Augmented Matched Interface and Boundary (AMIB) method for non-staggered boundaries. Formulated through immersed boundary problems with zero-padding solutions, the AMIB method combines arbitrarily high order central differences with the FFT inversion. Over staggered boundaries, the proposed AMIB method can handle Dirichlet, Neumann, Robin or any combination of boundary conditions. Convergence orders in four, six and eight are numerically validated for the AMIB method in both two and three dimensions. Moreover, the proposed AMIB method performs well for some challenging problems, such as low regularity solution near boundary, PDE solution not satisfying the boundary condition, and involving both staggered and non-staggered boundaries on two ends.