Filtered Hyperbolic Moment Method for the Vlasov Equation

In this paper, we investigate the effect of the filter for the hyperbolic moment equations (HME) (Cai et al. in Commun Pure Appl Math 67(3):464–518, 2014; Cai et al. in SIAM J Sci Comput 35(6):A2807–A2831, 2013) of the Vlasov–Poisson equations and propose a novel quasi time-consistent filter to suppress the numerical recurrence effect. By taking properties of HME into consideration, the filter preserves a lot of physical properties of HME, including Galilean invariance and conservation of mass, momentum and energy. We present two viewpoints—collisional viewpoint and dissipative viewpoint—to dissect the filter, and show that the filtered hyperbolic moment method can be treated as a solver of the Vlasov equation. Numerical simulations of the linear Landau damping and two stream instability demonstrate the effectiveness of the filter in restraining recurrence arising from particle streaming. Both the analysis and the numerical results indicate that the filtered method can capture the evolution of the Vlasov equation, even when phase mixing and filamentation dominate.

[1]  Manuel Torrilhon,et al.  Two-Dimensional Bulk Microflow Simulations Based on Regularized Grad's 13-Moment Equations , 2006, Multiscale Model. Simul..

[2]  Ruo Li,et al.  13-Moment System with Global Hyperbolicity for Quantum Gas , 2016, 1607.00175.

[3]  Zhenning Cai,et al.  Suppression of Recurrence in the Hermite-Spectral Method for Transport Equations , 2018, SIAM J. Numer. Anal..

[4]  High order moment closure for Vlasov-Maxwell equations , 2015 .

[5]  M. Shoucri,et al.  Numerical integration of the Vlasov equation , 1974 .

[6]  Paul J. Dellar,et al.  Fourier–Hermite spectral representation for the Vlasov–Poisson system in the weakly collisional limit , 2014, Journal of Plasma Physics.

[7]  Tiao Lu,et al.  Quantum Hydrodynamic Model by Moment Closure of Wigner Equation , 2012 .

[8]  Gian Luca Delzanno,et al.  On the velocity space discretization for the Vlasov-Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods , 2016, Comput. Phys. Commun..

[9]  Zhonghua Qiao,et al.  Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision , 2014 .

[10]  Eric Sonnendrücker,et al.  The Semi-Lagrangian Method for the Numerical Resolution of Vlasov Equations , 1998 .

[11]  E. Sonnendrücker,et al.  The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation , 1999 .

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

[13]  Ruo Li,et al.  Globally Hyperbolic Regularization of Grad's Moment System , 2012 .

[14]  Solène Le Bourdiec,et al.  Numerical solution of the Vlasov-Poisson system using generalized Hermite functions , 2006, Comput. Phys. Commun..

[15]  Ruo Li,et al.  Computing nearly singular solutions using pseudo-spectral methods , 2007, J. Comput. Phys..

[16]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[17]  P. J. Morrison,et al.  A discontinuous Galerkin method for the Vlasov-Poisson system , 2010, J. Comput. Phys..

[18]  Yingda Cheng,et al.  Study of conservation and recurrence of Runge–Kutta discontinuous Galerkin schemes for Vlasov–Poisson systems , 2012, J. Sci. Comput..

[19]  Joseph Oliger,et al.  Stability of the Fourier method , 1977 .

[20]  Ruo Li,et al.  Globally Hyperbolic Regularization of Grad's Moment System , 2011, 1203.0376.

[21]  L. Gardner,et al.  A finite element code for the simulation of one-dimensional Vlasov plasmas I. Theory , 1988 .

[22]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[23]  Emad Fatemi,et al.  Upwind Finite Difference Solution of Boltzmann Equation Applied to Electron Transport in Semiconductor Devices , 1993 .

[24]  Ryan G. McClarren,et al.  Robust and accurate filtered spherical harmonics expansions for radiative transfer , 2010, J. Comput. Phys..

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

[26]  Chi-Wang Shu,et al.  Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system , 2011, J. Comput. Phys..

[27]  Jan S. Hesthaven,et al.  Idempotent filtering in spectral and spectral element methods , 2006, J. Comput. Phys..

[28]  P. Bertrand,et al.  Conservative numerical schemes for the Vlasov equation , 2001 .

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

[30]  T. Armstrong,et al.  Numerical Studies of the Nonlinear Vlasov Equation , 1967 .

[31]  Ingo Müller,et al.  Waves in Extended Thermodynamics , 1998 .

[32]  Yuwei Fan,et al.  On Hyperbolicity of 13-Moment System , 2014, 1401.7523.

[33]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[34]  Ruo Li,et al.  Numerical Regularized Moment Method of Arbitrary Order for Boltzmann-BGK Equation , 2010, SIAM J. Sci. Comput..

[35]  B. Eliasson.  Numerical Simulations of the Fourier-Transformed Vlasov-Maxwell System in Higher Dimensions—Theory and Applications , 2010, 1007.4700.

[36]  Francis Filbet,et al.  Convergence of a Finite Volume Scheme for the Vlasov-Poisson System , 2001, SIAM J. Numer. Anal..

[37]  D. Gottlieb,et al.  Spectral methods for hyperbolic problems , 2001 .

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

[39]  Ruo Li,et al.  Solving Vlasov Equations Using NRxx Method , 2012, SIAM J. Sci. Comput..

[40]  Robert Michael Kirby,et al.  Filtering in Legendre spectral methods , 2008, Math. Comput..

[41]  Alexander J. Klimas,et al.  A method for overcoming the velocity space filamentation problem in collisionless plasma model solutions , 1987 .

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

[43]  Francis Filbet,et al.  Numerical approximation of collisional plasmas by high order methods , 2004 .

[44]  Marc R. Feix,et al.  Fourier‐Hermite Solutions of the Vlasov Equations in the Linearized Limit , 1967 .

[45]  Bengt Eliasson.  Outflow Boundary Conditions for the Fourier Transformed One-Dimensional Vlasov–Poisson System , 2001, J. Sci. Comput..

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

[47]  Irene M. Gamba,et al.  A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods , 2003 .

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

[49]  E. Sonnendrücker,et al.  Comparison of Eulerian Vlasov solvers , 2003 .

[50]  M. Feix,et al.  TRANSITION BETWEEN LANDAU AND VAN KAMPEN TREATMENTS OF THE VLASOV EQUATION. , 1967 .

[51]  Joseph E. Flaherty,et al.  A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations , 1986 .

[52]  Lev Davidovich Landau,et al.  On the vibrations of the electronic plasma , 1946 .

[53]  R. Contreras,et al.  π‐Strain‐Induced Electrophilicity in Small Cycloalkynes: A DFT Analysis of the Polar Cycloaddition of Cyclopentyne towards Enol Ethers , 2006 .