American Institute of Mathematical Sciences

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

Accurate two and three dimensional interpolation for particle mesh calculations

Anita Mayo 1,

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.
Keywords: Particle Mesh method, interpolation..
Mathematics Subject Classification: Primary: 65D05; Secondary: 65Z0.
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.

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