A high order solver for the unbounded Poisson equation

A high order converging Poisson solver is presented, based on the Green?s function solution to Poisson?s equation subject to free-space boundary conditions. The high order convergence is achieved by formulating regularised integration kernels, analogous to a smoothing of the solution field. The method is extended to directly solve the derivatives of the solution to Poisson?s equation. In this way differential operators such as the divergence or curl of the solution field can be solved to the same high order convergence without additional computational effort. The method, is applied and validated, however not restricted, to the equations of fluid mechanics, and can be used in many applications to solve Poisson?s equation on a rectangular unbounded domain.

[1]  Reuben D. Budiardja,et al.  Parallel FFT-based Poisson solver for isolated three-dimensional systems , 2011, Comput. Phys. Commun..

[2]  Grégoire Winckelmans,et al.  Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows , 1993 .

[3]  D. Lieberman,et al.  Fourier analysis , 2004, Journal of cataract and refractive surgery.

[4]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[5]  Jens Honore Walther,et al.  Particle Methods in Bluff Body Aerodynamics , 2011 .

[6]  A. Leonard.  Vortex methods for flow simulation , 1980 .

[7]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[8]  Michael Griebel,et al.  Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications , 2007 .

[9]  Ole H. Hald,et al.  Convergence of Vortex methods for Euler's equations, III , 1987 .

[10]  Georges-Henri Cottet,et al.  A multiresolution remeshed Vortex-In-Cell algorithm using patches , 2011, J. Comput. Phys..

[11]  Petros Koumoutsakos,et al.  Vortex Methods: Theory and Practice , 2000 .

[12]  J. CARRIERt,et al.  A FAST ADAPTIVE MULTIPOLE ALGORITHM FOR PARTICLE SIMULATIONS * , 2022 .

[13]  Guido Morgenthal,et al.  An immersed interface method for the vortex-in-cell algorithm , 2007 .

[14]  T. Colonius,et al.  A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions , 2008 .

[15]  Petros Koumoutsakos,et al.  A Fourier-based elliptic solver for vortical flows with periodic and unbounded directions , 2010, J. Comput. Phys..

[16]  A. Majda,et al.  Vortex methods. II. Higher order accuracy in two and three dimensions , 1982 .

[17]  P A Raviart.  Particle approximation of first order systems , 1986 .

[18]  Marjorie Perlman,et al.  Accuracy of vortex methods , 1983 .

[19]  Ole Hald,et al.  Convergence of vortex methods for Euler’s equations , 1978 .

[20]  Christopher R. Anderson,et al.  On Vortex Methods , 1985 .

[21]  Andrew J. Majda,et al.  High order accurate vortex methods with explicit velocity kernels , 1985 .

[22]  Philippe Chatelain,et al.  Isotropic compact interpolation schemes for particle methods , 2008, J. Comput. Phys..

[23]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[24]  Phillip Colella,et al.  Advanced 3D Poisson solvers and particle-in-cell methods for accelerator modeling , 2005 .

[25]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[26]  Andrew J. Majda,et al.  Vortex Methods. I: Convergence in Three Dimensions , 2010 .

[27]  Ji Qiang.  A high-order fast method for computing convolution integral with smooth kernel , 2010, Comput. Phys. Commun..

[28]  R. A. James,et al.  The solution of Poisson''s equation for isolated source distributions , 1977 .

[29]  J. Christiansen.  Numerical Simulation of Hydrodynamics by the Method of Point Vortices , 1997 .

[30]  Jiro Shimada,et al.  Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations , 1994, J. Comput. Chem..

[31]  Grégoire Winckelmans,et al.  Combining the vortex-in-cell and parallel fast multipole methods for efficient domain decomposition simulations , 2008, J. Comput. Phys..