A Spectral Algorithm for Solving the Relativistic Vlasov-Maxwell Equations

A spectral method algorithm is developed for the numerical solution of the full six-dimensional Vlasov-Maxwell system of equations. Here, the focus is on the electron distribution function, with positive ions providing a constant background. The algorithm consists of a Jacobi polynomial-spherical harmonic formulation in velocity space and a trigonometric formulation in position space. A transform procedure is used to evaluate nonlinear terms. The algorithm is suitable for performing moderate resolution simulations on currently available supercomputers for both scientific and engineering applications.

[1]  A. Klimas,et al.  A splitting algorithm for Vlasov simulation with filamentation filtration , 1994 .

[2]  J. Shebalin.  Electromagnetic potential vectors and spontaneous symmetry breaking , 1993 .

[3]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[4]  Tomoaki Kunugi,et al.  A numerical method for solving the one-dimensional Vlasov—Poisson equation in phase space , 1998 .

[5]  Réal R. J. Gagné,et al.  A Splitting Scheme for the Numerical Solution of a One-Dimensional Vlasov Equation , 1977 .

[6]  M. Brereton.  Classical Electrodynamics (2nd edn) , 1976 .

[7]  J. Gazdag.  Time-differencing schemes and transform methods , 1976 .

[8]  A. Krall.  Applied Analysis , 1986 .

[9]  G. Joyce,et al.  Numerical integration methods of the Vlasov equation , 1971 .

[10]  Jacques Denavit,et al.  Numerical simulation of plasmas with periodic smoothing in phase space , 1972 .

[11]  F. Yuan,et al.  SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) , 1999 .

[12]  G. Arfken,et al.  Mathematical methods for physicists 6th ed. , 1996 .

[13]  G. S. Patterson,et al.  Spectral Calculations of Isotropic Turbulence: Efficient Removal of Aliasing Interactions , 1971 .

[14]  G. Knorr,et al.  The integration of the vlasov equation in configuration space , 1976 .

[15]  Joseph W. Schumer,et al.  Vlasov Simulations Using Velocity-Scaled Hermite Representations , 1998 .

[16]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[17]  G. Edwards.  Texas , 1958, "These United States".

[18]  James Paul Holloway,et al.  Spectral velocity discretizations for the Vlasov-Maxwell equations , 1996 .

[19]  M. Shoucri,et al.  Stability of Bernstein–Greene–Kruskal plasma equilibria. Numerical experiments over a long time , 1988 .

[20]  Homogeneous quantum electrodynamic turbulence , 1992 .

[21]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[22]  T. Armstrong,et al.  Numerical Study of Weakly Unstable Electron Plasma Oscillations , 1969 .

[23]  J. Denavit.  Simulations of the single‐mode, bump‐on‐tail instability , 1985 .

[24]  B. Buti.  Plasma Oscillations and Landau Damping in a Relativistic Gas , 1962 .

[25]  John V. Shebalin,et al.  Numerical solution of the coupled Dirac and Maxwell equations , 1997 .

[26]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[27]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[28]  T. Armstrong,et al.  Asymptotic state of the two-stream instability , 1967, Journal of Plasma Physics.

[29]  S. Orszag.  Transform method for the calculation of vector-coupled sums: Application to the spectral form of the vorticity equation , 1970 .

[30]  Jacques Denavit,et al.  Comparison of Numerical Solutions of the Vlasov Equation with Particle Simulations of Collisionless Plasmas , 1971 .

[31]  R. C. Harding,et al.  SOLUTION OF VLASOV'S EQUATION BY TRANSFORM METHODS. , 1970 .

[32]  G. Joyce,et al.  Nonlinear Behavior of the One‐Dimensional Weak Beam Plasma System , 1971 .

[33]  G. Knorr.  Plasma simulation with few particles. , 1973 .

[34]  J. Holloway.  On numerical methods for Hamiltonian PDEs and a collocation method for the Vlasov-Maxwell equations , 1996 .

[35]  J. Denavit,et al.  Nonlinear and Collisional Effects on Landau Damping , 1968 .

[36]  T. Yabe,et al.  Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space , 1999 .

[37]  Stephen Wollman,et al.  Numerical Approximation of the One-Dimensional Vlasov--Poisson System with Periodic Boundary Conditions , 1996 .

[38]  The inclusion of collisional effects in the splitting scheme , 1992 .

[39]  H. Qin,et al.  Kinetic description of electron-proton instability in high-intensity proton linacs and storage rings based on the Vlasov-Maxwell equations , 1999 .

[40]  P. Clemmow,et al.  The dispersion equation in plasma oscillations , 1956, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[41]  E. M. Lifshitz,et al.  Classical theory of fields , 1952 .