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Structure preservation using discrete gradients in the Vlasov-Poisson-Landau system

Daniel S. Finn, Joseph V. Pusztay, Matthew G. Knepley, Mark F. Adams

Abstract

We present a novel structure-preserving framework for solving the Vlasov-Poisson-Landau system of equations using a particle in cell (PIC) discretization combined with discrete gradient time integrators. The Vlasov-Poisson-Landau system is an accurate model for studying hot plasma dynamics at a kinetic scale where small-angle Coulomb collisions dominate. Our scheme guarantees conservation of mass, momentum and energy as well as preservation of the monotonicity of entropy production in both the time-continuous and discrete systems. We employ the conservative integrator for both the Hamiltonian Vlasov-Poisson equations and the dissipative Landau equation using the PETSc library (www.mcs.anl.gov/petsc) to showcase structure-preserving properties.

Structure preservation using discrete gradients in the Vlasov-Poisson-Landau system

Abstract

We present a novel structure-preserving framework for solving the Vlasov-Poisson-Landau system of equations using a particle in cell (PIC) discretization combined with discrete gradient time integrators. The Vlasov-Poisson-Landau system is an accurate model for studying hot plasma dynamics at a kinetic scale where small-angle Coulomb collisions dominate. Our scheme guarantees conservation of mass, momentum and energy as well as preservation of the monotonicity of entropy production in both the time-continuous and discrete systems. We employ the conservative integrator for both the Hamiltonian Vlasov-Poisson equations and the dissipative Landau equation using the PETSc library (www.mcs.anl.gov/petsc) to showcase structure-preserving properties.
Paper Structure (24 sections, 1 theorem, 119 equations, 6 figures)

This paper contains 24 sections, 1 theorem, 119 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Vlasov-Poisson-Landau System
  3. Bracket Formulation
  4. Poisson Bracket
  5. Finite Element Discretization of the Poisson Bracket
  6. Metric Bracket
  7. Marker-Particle Discretization of the Collisional Bracket
  8. Temporal Discretization
  9. Discrete Gradient Discretization of the Poisson Bracket
  10. Discrete Gradient Dependent Integrators for the Metric Bracket
  11. Numerical Implementation
  12. Solver Considerations
  13. Collisionless Dynamics
  14. Collisional Dynamics
  15. Conclusions
  16. ...and 9 more sections

Key Result

Theorem 4.1

The collisional discrete-time evolution equation, given by DGColDTE, is a consistent approximation to the finite-dimensional bracket FinDimBrack. in the limit where $||\boldsymbol{v}_{p}^{n+1} - \boldsymbol{v}_{p}^{n}|| \rightarrow 0$ and $||\boldsymbol{v}_{\overline{p}}^{n+1} - \boldsymbol{v}_{\overline{p}}^{n}|| \rightarrow 0$.

Figures (6)

  • Figure 1: The max-norm of the electric field as a function of time for three different timestepping integrators: discrete gradient integrators, 1st-order symplectic, and 4th-order Runge-Kutta. For this test, the relative tolerance for the discrete gradient methods was set to $10^{-11}$.
  • Figure 2: The moments, mass, momentum and energy, as a function of time for three different timestepping integrators. Absolute error is used for the momentum plot as the true value of momentum is zero, and therefore, the relative error is non-convergent.
  • Figure 3: Entropy and regularized entropy for each of the three different timestepping integrators. Regularized entropy is only computed every 100 steps as it is extremely computationally expensive ($\mathcal{O} (N^2)$).
  • Figure 4: A comparison of energy conservation for two nonlinear solver types: quasi-Newton (L-BFGS) and nonlinear Richardson (fixed-point). Each solver was tested with two relative tolerance levels. The nonlinear Richardson solver was given a larger tolerance difference as the results showed no change with the original tolerance difference.
  • Figure 5: (Top) Electron-positron equilibration using a non-conservative Euler and the average discrete gradient time integrator for the particle basis Landau collision integral compared with analytical solution. (Middle) Absolute error in momentum for electron-positron equilibration for each time integrator.(Bottom) Relative error in total energy for electron-positron equilibration for each time integrator.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 4.1
  • Definition 4.2
  • Theorem 4.1
  • proof