A Cartesian treecode for screened coulomb interactions

A treecode algorithm is presented for evaluating electrostatic potentials in a charged particle system undergoing screened Coulomb interactions in 3D. The method uses a far-field Taylor expansion in Cartesian coordinates to compute particle-cluster interactions. The Taylor coefficients are evaluated using new recurrence relations which permit efficient computation of high order approximations. Two types of clusters are considered, uniform cubes and adapted rectangular boxes. The treecode error, CPU time and memory usage are reported and compared with direct summation for randomly distributed particles inside a cube, on the surface of a sphere and on an 8-sphere configuration. For a given order of Taylor approximation, the treecode CPU time scales as O ( N log N ) and the memory usage scales as O ( N ) , where N is the number of particles. Results show that the treecode is well suited for non-homogeneous particle distributions as in the sphere and 8-sphere test cases.

[1]  A. Piel,et al.  Structural properties of screened Coulomb balls. , 2005, Physical review letters.

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

[3]  Tamar Schlick,et al.  Molecular Modeling and Simulation: An Interdisciplinary Guide , 2010 .

[4]  Nathan A. Baker,et al.  Improving implicit solvent simulations: a Poisson-centric view. , 2005, Current opinion in structural biology.

[5]  Robert Krasny,et al.  An Ewald summation based multipole method , 2000 .

[6]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .

[7]  Z. Duan,et al.  An adaptive treecode for computing nonbonded potential energy in classical molecular systems , 2001 .

[8]  W. Goddard,et al.  Atomic level simulations on a million particles: The cell multipole method for Coulomb and London nonbond interactions , 1992 .

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

[10]  Benzhuo Lu,et al.  Order N algorithm for computation of electrostatic interactions in biomolecular systems , 2006, Proceedings of the National Academy of Sciences.

[11]  C. I. Draghicescu,et al.  A fast algorithm for vortex blob interactions , 1995 .

[12]  B. Honig,et al.  Classical electrostatics in biology and chemistry. , 1995, Science.

[13]  W. Theisen,et al.  Study of two-dimensional Debye clusters using Brownian motion , 2006 .

[14]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[15]  K. Lindsay,et al.  A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow , 2001 .

[16]  Marcia O. Fenley,et al.  Fast Boundary Element Method for the Linear Poisson-Boltzmann Equation , 2002 .

[17]  Walter Dehnen,et al.  A Hierarchical O(N) Force Calculation Algorithm , 2002 .

[18]  L. Greengard,et al.  A new version of the fast multipole method for screened Coulomb interactions in three dimensions , 2002 .

[19]  Takashi Sakajo,et al.  An Application of Draghicescu's Fast Summation Method to Vortex Sheet Motion , 1998 .

[20]  J. A. McCammon,et al.  R EVIEW A RTICLE Recent Progress inNumericalMethods for the Poisson-Boltzmann Equation in Biophysical Applications , 2008 .

[21]  He Huang,et al.  Accelerated Cartesian expansions - A fast method for computing of potentials of the form R-ν for all real ν , 2007, J. Comput. Phys..

[22]  D. Zorin,et al.  A kernel-independent adaptive fast multipole algorithm in two and three dimensions , 2004 .

[23]  Konopka,et al.  Measurement of the interaction potential of microspheres in the sheath of a rf discharge , 2000, Physical review letters.

[24]  Marcia O. Fenley,et al.  A Fast Adaptive Multipole Algorithm for Calculating Screened Coulomb (Yukawa) Interactions , 1999 .

[25]  Andrew D. Jackson,et al.  Nucleon--nucleon interaction , 1974 .

[26]  Benzhuo Lu,et al.  "New-version-fast-multipole-method" accelerated electrostatic calculations in biomolecular systems , 2007, J. Comput. Phys..

[27]  T Schlick,et al.  The influence of salt on the structure and energetics of supercoiled DNA. , 1994, Biophysical journal.

[28]  Michael S. Warren,et al.  Skeletons from the treecode closet , 1994 .

[29]  Feng Zhao,et al.  An {\it bf O(N)} Algorithm for Three-Dimensional N-body Simulations , 1987 .

[30]  L. Greengard.  The Rapid Evaluation of Potential Fields in Particle Systems , 1988 .

[31]  Feng Zhao.  An O(N) Algorithm for Three-dimensional N-body Simulations , 2022 .