A High-Order Fast Direct Solver for Singular Poisson Equations

Abstract

We present a fourth order numerical solution method for the singular Neumann boundary problem of Poisson equations. Such problems arise in the solution process of incompressible Navier-Stokes equations and in the time-harmonic wave propagation in the frequence space with the zero wavenumber. The equation is first discretized with a fourth order modified Collatz difference scheme, producing a singular discrete equation. Then an efficient singular value decomposition (SVD) method modified from a fast Poisson solver is employed to project the discrete singular equation into the orthogonal complement of the null space of the singular matrix. In the complement of the null space, the projected equation is uniquely solvable and its solution is proven to be a solution of the original singular discrete equation when the original equation has a solution. Analytical and experimental results show that this newly proposed singular equation solver is efficient while retaining the accuracy of the high order discretization.

A Discrete Isoperimetric Inequality on Lattices

“…One is a solvability of the discrete PoissonNeumann problem. Such problems are discussed in the previous work [26,23,25,28], but domains are restricted to rectangles ( [26,23,28]) or their collections ( [25]). For the proof of our discrete isoperimetric inequality, fortunately, it is enough to require u to be a subsolution of the Poisson equation in (1.4) and to satisfy the Neumann condition in (1.4) with some direction ν.…”

Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations

“…The Poisson equations for velocity in Eqs. (38) and (39) are discretized with forth-order accuracy proposed by Zhuang and Sun [24], and the Gauss-Seidel iterative method is applied to solve the discrete Poisson equations with the infinite-norm error during two iterative steps smaller than 10 À5 . To validate present immersed boundary method based on vorticity-velocity formulations, three test cases including decaying vortices, flow past a stationary circular cylinder and an in-line oscillating cylinder in a fluid at rest are conducted.…”

A transformation-free HOC scheme for incompressible viscous flows past an impulsively started circular cylinder

“…The basic difference between the proposed scheme and the earlier HOC schemes is that the present scheme is able to handle variable coefficients of the second order derivatives while the previous schemes could deal with unit diffusion coefficients only. This perhaps is the reason that majority of the earlier endeavors to develop HOC schemes on cylindrical polar coordinates were confined to the Poisson equation on uniform grids [48][49][50][51][52] only. To validate the proposed scheme, we apply it to this well known problem of unsteady flow past an impulsively started circular cylinder for a wide range of Re ranging from 10 to 9500.…”