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A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas

L. Chacon, Jason Hamilton, Natalia Krasheninnikova

Abstract

We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp. Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a second-order-accurate approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.

A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas

Abstract

We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp. Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a second-order-accurate approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.
Paper Structure (13 sections, 28 equations, 12 figures)

This paper contains 13 sections, 28 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model equations, numerical discretization, and solver strategy
  3. Fourth-order discretization of the anisotropic diffusion equation
  4. Reformulation and discretization of cross-derivative diffusion fluxes
  5. Temporal discretization, nonlinear solver strategy, and multigrid preconditioning
  6. Numerical tests
  7. NIMROD benchmark test
  8. Magnetic island test
  9. Bennett screw-pinch kink instability MHD simulation in 2D helical geometry
  10. ITER kink instability MHD simulation in 3D toroidal geometry
  11. Conclusions
  12. Implementation details of the fourth-order accurate discretization
  13. Logical-to-physical coordinate map in the ITER simulation

Figures (12)

  • Figure 1: Combined stencil for discretization of parallel transport operator. Black points correspond to co-derivative fluxes, and gray points to cross-derivative ones.
  • Figure 2: NIMROD benchmark test: relative error $\Delta\chi$ at steady state for second-order and fourth-order discretizations with various grid resolutions and $\epsilon=10^{-3},10^{-5}$. Expected asymptotic convergence rates are recovered by the scheme.
  • Figure 3: NIMROD benchmark test: scaling of $\Delta\chi$ with the anisotropy $\epsilon$.
  • Figure 4: NIMROD benchmark test: number of GMRES iterations per time step vs. $\epsilon$ for the fourth-order scheme for (a) different mesh resolutions for $\Delta t\chi_{\parallel}=1$, and (b) various $\Delta t\chi_{\parallel}$ for the $64\times64$ mesh.
  • Figure 5: Magnetic island test: steady-state solution (t = 0.62) in Cartesian coordinates for $\chi_{\perp}/\chi_{\parallel}=10^{-7}$ on a 512x256 mesh. The magnetic island separatrix is indicated with a black line.
  • ...and 7 more figures