An FFT Based Fast Poisson Solver on Spherical Shells

Revista Brasileira de Ensino de Física, 2021

This paper sets out to present a numerical procedure that solves Poisson's equation in a spherical coordinate system. To discretize this equation, integration techniques at the interfaces between different regions have been carried out allowing the calculation of both the potential and the corresponding field inside and outside a charge distribution. The Gauss-Seidel method is adopted to determine the potential in each region and the results, whenever compared with the analytical solutions found in the literature, come out very satisfactory, with errors less than 1% for distances of the order of 1 × 10 −14 m and, for larger distances, they never reach 4%.

Communications in Computational Physics, 2022

Poisson's equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson's equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green's function within a sphere containing the cuboid, and another surface integration over the spherical surface. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algorithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.

We present an adaptive fast multipole method for solving the Poisson equation in two dimensions. The algorithm is direct, assumes that the source distribution is discretized using an adaptive quad-tree, and allows for Dirichlet, Neumann, periodic, and free-space conditions to be imposed on the boundary of a square. The amount of work per grid point is comparable to that of classical fast solvers, even for highly nonuniform grids.

Quarterly Journal of the Royal Meteorological Society, 2009

ABSTRACT Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude–longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost.The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourier-finite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N2 degrees of freedom, the computational cost is (N2logN), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs (N2), and less than with a spherical harmonics method, which costs (N3). Differential operators with latitude-dependent coefficients are inverted at a cost of (N2).We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier–Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described. Copyright © 2009 Royal Meteorological Society

2001

We present an adaptive fast multipole method for solving the Poisson equation in two dimensions. The algorithm is direct, assumes that the source distribution is discretized using an adaptive quad-tree, and allows for Dirichlet, Neumann, periodic, and free-space conditions to be imposed on the boundary of a square. The amount of work per grid point is comparable to that of classical fast solvers, even for highly nonuniform grids.

IMA Journal of Numerical Analysis, 2019

Poisson’s equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference (FD) and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here we derive spectral methods for solving Poisson’s equation on a square, cylinder, solid sphere and cube that have optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom used to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of FD matrices, our solver exploits a separated spectra property that holds for our carefully designed spectral discretizations. Without parallelization we can solve Poisson’s equation on a square with 100 million degrees of freedom in under 2 min on a standard laptop.

Journal of King Abdulaziz University-Science, 2003

Fast algorithm for the accurate evaluation of some integral operator that arise in the context of solving certain partial differential equations within the unit circle in the complex plane are presented. It is based on some recursive relations in the Fourier space and the FFT (Fast Fourier Transform), and have theoretical computational complexity of the order O(log N) per point, where N 2 is the total number of grid points.

2014

We present a fast and accurate algorithm to solve Poisson problems in complex geometries, using regular Cartesian grids. We consider a variety of configurations, including Poisson problems with interfaces across which the solution is discontinuous (of the type arising in multi-fluid flows). The algorithm is based on a combination of the Correction Function Method (CFM) and Boundary Integral Methods (BIM). Interface and boundary conditions can be treated in a fast and accurate manner using boundary integral equations, and the associated BIM. Unfortunately, BIM can be costly when the solution is needed everywhere in a grid, e.g. fluid flow problems. We use the CFM to circumvent this issue. The solution from the BIM is used to rewrite the problem as a series of Poisson problems in rectangular domains - which requires the BIM solution at interfaces/boundaries only. These Poisson problems involve discontinuities at interfaces, of the type that the CFM can handle. Hence we use the CFM to solve them (to high order of accuracy) with finite differences and a Fast Fourier Transform based fast Poisson solver. We present 2-D examples of the algorithm applied to Poisson problems involving complex geometries, including cases in which the solution is discontinuous. We show that the algorithm produces solutions that converge with either 3rd or 4th order of accuracy, depending on the type of boundary condition and solution discontinuity.

SIAM Journal on Scientific Computing, 2013

The fast multipole method (FMM) has had great success in reducing the computational time required to solve the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM using Fourier basis functions rather than spherical harmonics that accelerates some of the time-critical stages of the algorithm. With modifications to the transfer function in the precomputation stage of the FMM, the interpolation and anterpolation operators become straightforward applications of fast Fourier transforms and the transfer operator remains diagonal. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function.

International Journal for Numerical Methods in Fluids, 2005

A fast cosine transform (FCT) is coupled with a tridiagonal solver for the purpose of solving the Poisson equation on irregular and non-uniform rectangular staggered grids. This kind of solution is required for the pressure ÿeld during the simulation of the incompressible Navier-Stokes equations when using the projection method. A new technique using the FCT-tridiagonal solver is derived for the cases where the boundaries of the ow regime do not coincide with the boundaries of the computational domain and for non-uniform grids. The technique is based on an iterative procedure where a defect equation is solved in every iteration, followed by a relaxation procedure. The method is investigated analytically and numerically to show that the solution converges as a geometric series. The method is further investigated for the e ects of the relative size of the rigid body, the grid stretching, size and aspect ratio. The new solver is incorporated with the direct numerical simulation (DNS) and large eddy simulation (LES) techniques to simulate the ows around a backward-facing step and a 3D rectangular obstacle, yielding results that qualitatively compare well with known results. Copyright ? 2005 John Wiley & Sons, Ltd.