The energy conserving particle-in-cell method

Abstract A new particle-in-cell (PIC) method, that conserves energy exactly, is presented. The particle equations of motion and the Maxwell’s equations are differenced implicitly in time by the midpoint rule and solved concurrently by a Jacobian-free Newton Krylov (JFNK) solver. Several tests show that the finite grid instability is eliminated in energy conserving PIC simulations, and the method correctly describes the two-stream and Weibel instabilities, conserving exactly the total energy. The computational time of the energy conserving PIC method increases linearly with the number of particles, and it is rather insensitive to the number of grid points and time step. The kinetic enslavement technique can be effectively used to reduce the problem matrix size and the number of JFNK solver iterations.

[1]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[2]  E. S. Weibel,et al.  Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution , 1959 .

[3]  J. Dawson.  Particle simulation of plasmas , 1983 .

[4]  Luis Chacon,et al.  Fully implicit particle-in-cell-algorithm , 2005 .

[5]  B. M. Marder,et al.  A method for incorporating Gauss' lasw into electromagnetic pic codes , 1987 .

[6]  M. Shashkov,et al.  Mimetic Discretizations for Maxwell's Equations , 1999 .

[7]  A. Arakawa,et al.  Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations , 1990 .

[8]  Y. Omura,et al.  Particle simulation of electromagnetic waves and its application to space plasmas. , 1985 .

[9]  C. Goertz.  Computer Simulations of Space Plasmas , 1987 .

[10]  E. Lindman,et al.  DISPERSION RELATION FOR COMPUTER-SIMULATED PLASMAS. , 1970 .

[11]  Kwang-Ting Cheng,et al.  Fundamentals of algorithms , 2009 .

[12]  Jeremiah Brackbill,et al.  Simulation of Low-Frequency, Electromagnetic Phenomena in Plasmas , 1985 .

[13]  H. Lewis,et al.  Energy-conserving numerical approximations for Vlasov plasmas , 1970 .

[14]  C. Birdsall,et al.  Plasma Physics via Computer Simulation , 2018 .

[15]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[16]  J. Brackbill,et al.  Multiple time scales , 1985 .

[17]  A. Bruce Langdon.  Some Electromagnetic Plasma Simulation Methods and Their Noise Properties , 1972 .

[18]  Stefano Markidis,et al.  Development of implicit kinetic simulation methods, and their application to ion beam propagation in current and future neutralized drift compression experiments , 2010 .

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

[20]  A. Bruce Langdon,et al.  “Energy-conserving” plasma simulation algorithms , 1973 .

[21]  C. T. Kelley,et al.  Solving nonlinear equations with Newton's method - fundamentals of algorithms , 2003 .

[22]  A. Bruce Langdon.  Analysis of the time integration in plasma simulation , 1979 .

[23]  N. A. Krall,et al.  Principles of Plasma Physics , 1973 .

[24]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[25]  R. Morse,et al.  NUMERICAL SIMULATION OF THE WEIBEL INSTABILITY IN ONE AND TWO DIMENSIONS. , 1971 .

[26]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[27]  K. Yee.  Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media , 1966 .

[28]  A. Bruce Langdon,et al.  On enforcing Gauss' law in electromagnetic particle-in-cell codes , 1992 .