June  2012, 17(4): 1205-1228. doi: 10.3934/dcdsb.2012.17.1205

Accurate two and three dimensional interpolation for particle mesh calculations

1. 

Mathematics Department, Baruch College CUNY, New York, NY, United States

Received  January 2011 Revised  December 2011 Published  February 2012

We present accurate two and three dimensional methods for interpolating singular or smoothed force fields. The methods are meant to be used in particle mesh or particle-particle particle-mesh calculations so that the resulting schemes conserve momentum. The interpolation weights, which have previously been used by Anderson and Colella to spread charge from particles to the mesh (but not to interpolate the force from the mesh to the particles) use discretizations of the differential equations the forces satisfy. The methods are most accurate when the forces satisfy homogeneous elliptic differential equations or systems of equations, and the precise accuracy levels of the interpolation formulas depend on the accuracy of certain corresponding quadrature formulas. We describe the methods and give results of numerical experiments which demonstrate their effectiveness.
Citation: Anita Mayo. Accurate two and three dimensional interpolation for particle mesh calculations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1205-1228. doi: 10.3934/dcdsb.2012.17.1205
References:
[1]

C. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. Comp. Phys., 62 (1986), 111-123. doi: 10.1016/0021-9991(86)90102-6.

[2]

K. Atkinson, "An Introduction to Numerical Analysis," John Wiley & Sons, New York-Chichester-Brisbane, 1978.

[3]

C. K. Birdsall and W. Bridges, Space-charge instabilities in electron diodes and plasma converters, J. Appl. Phys., 32 (1961), 2611-2618. doi: 10.1063/1.1728361.

[4]

O. Buneman, Dissipation of currents in ionised media, Phys. Rev., 115 (1959), 503-517. doi: 10.1103/PhysRev.115.503.

[5]

P. Colella and P. Norgaard, Controlling self-force errors at refinement boundaries for AMR-PIC, J. Comp. Phys., 299 (2010), 947-957. doi: 10.1016/j.jcp.2009.07.004.

[6]

H. Engels, "Numerical Cubature and Quadrature," Computational Mathematics and Applications, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.

[7]

M. Gupta, R. Mandohar and J. Stephenson, High-order difference scheme for two dimensional elliptic equations, Numer. Methods Partial Differential Equations, 1 (1985), 71-80. doi: 10.1002/num.1690010108.

[8]

R. Hockney, R. Warriner and J. Reiset, Two dimensional particle models in semiconductor device analysis, Electron. Lett., 10 (1974), 484-486. doi: 10.1049/el:19740386.

[9]

R. Hockney and J. Eastwood, "Computer Simulation Using Particles," McGraw Hill, New York, 1979.

[10]

R. Lynch and J. Rice, A high-order difference method for differential equations, Math. Comp., 34 (1980), 333-372.

[11]

A. Mayo, Fast, high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. and Stat. Comput., 6 (1985), 144-157. doi: 10.1137/0906012.

[12]

C. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Phys, 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.

show all references

References:
[1]

C. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. Comp. Phys., 62 (1986), 111-123. doi: 10.1016/0021-9991(86)90102-6.

[2]

K. Atkinson, "An Introduction to Numerical Analysis," John Wiley & Sons, New York-Chichester-Brisbane, 1978.

[3]

C. K. Birdsall and W. Bridges, Space-charge instabilities in electron diodes and plasma converters, J. Appl. Phys., 32 (1961), 2611-2618. doi: 10.1063/1.1728361.

[4]

O. Buneman, Dissipation of currents in ionised media, Phys. Rev., 115 (1959), 503-517. doi: 10.1103/PhysRev.115.503.

[5]

P. Colella and P. Norgaard, Controlling self-force errors at refinement boundaries for AMR-PIC, J. Comp. Phys., 299 (2010), 947-957. doi: 10.1016/j.jcp.2009.07.004.

[6]

H. Engels, "Numerical Cubature and Quadrature," Computational Mathematics and Applications, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.

[7]

M. Gupta, R. Mandohar and J. Stephenson, High-order difference scheme for two dimensional elliptic equations, Numer. Methods Partial Differential Equations, 1 (1985), 71-80. doi: 10.1002/num.1690010108.

[8]

R. Hockney, R. Warriner and J. Reiset, Two dimensional particle models in semiconductor device analysis, Electron. Lett., 10 (1974), 484-486. doi: 10.1049/el:19740386.

[9]

R. Hockney and J. Eastwood, "Computer Simulation Using Particles," McGraw Hill, New York, 1979.

[10]

R. Lynch and J. Rice, A high-order difference method for differential equations, Math. Comp., 34 (1980), 333-372.

[11]

A. Mayo, Fast, high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. and Stat. Comput., 6 (1985), 144-157. doi: 10.1137/0906012.

[12]

C. Peskin, Numerical analysis of blood flow in the heart, J. Comp. Phys, 25 (1977), 220-252. doi: 10.1016/0021-9991(77)90100-0.

[1]

Rongjie Lai, Jiang Liang, Hong-Kai Zhao. A local mesh method for solving PDEs on point clouds. Inverse Problems and Imaging, 2013, 7 (3) : 737-755. doi: 10.3934/ipi.2013.7.737

[2]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[3]

José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic and Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025

[4]

Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1

[5]

Xiu Ye, Shangyou Zhang. A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh. Electronic Research Archive, 2021, 29 (6) : 3609-3627. doi: 10.3934/era.2021053

[6]

Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

[7]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial and Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115

[8]

Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 727-746. doi: 10.3934/dcdss.2021097

[9]

Xiaoxue Gong, Ying Xu, Vinay Mahadeo, Tulin Kaman, Johan Larsson, James Glimm. Mesh convergence for turbulent combustion. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4383-4402. doi: 10.3934/dcds.2016.36.4383

[10]

Vincent Ducrot, Pascal Frey, Alexandra Claisse. Levelsets and anisotropic mesh adaptation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 165-183. doi: 10.3934/dcds.2009.23.165

[11]

Jean Dolbeault, Maria J. Esteban, Michał Kowalczyk, Michael Loss. Improved interpolation inequalities on the sphere. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 695-724. doi: 10.3934/dcdss.2014.7.695

[12]

Charles Fefferman. Interpolation by linear programming I. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477

[13]

Birol Yüceoǧlu, ş. ilker Birbil, özgür Gürbüz. Dispersion with connectivity in wireless mesh networks. Journal of Industrial and Management Optimization, 2018, 14 (2) : 759-784. doi: 10.3934/jimo.2017074

[14]

Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175

[15]

Jean Dolbeault, An Zhang. Parabolic methods for ultraspherical interpolation inequalities. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022080

[16]

Omar Saber Qasim, Ahmed Entesar, Waleed Al-Hayani. Solving nonlinear differential equations using hybrid method between Lyapunov's artificial small parameter and continuous particle swarm optimization. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 633-644. doi: 10.3934/naco.2021001

[17]

Nahid Banihashemi, C. Yalçın Kaya. Inexact restoration and adaptive mesh refinement for optimal control. Journal of Industrial and Management Optimization, 2014, 10 (2) : 521-542. doi: 10.3934/jimo.2014.10.521

[18]

Yvon Maday, Ngoc Cuong Nguyen, Anthony T. Patera, S. H. Pau. A general multipurpose interpolation procedure: the magic points. Communications on Pure and Applied Analysis, 2009, 8 (1) : 383-404. doi: 10.3934/cpaa.2009.8.383

[19]

Tony Lyons. Particle paths in equatorial flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2399-2414. doi: 10.3934/cpaa.2022041

[20]

Vyacheslav K. Isaev, Vyacheslav V. Zolotukhin. Introduction to the theory of splines with an optimal mesh. Linear Chebyshev splines and applications. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 471-489. doi: 10.3934/naco.2013.3.471

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (692)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]