References and related work

This page collects the main literature that informs the physics model, numerics, validation strategy, and workflow design of dkx.

Foundational neoclassical theory

P. Helander and D. J. Sigmar, Collisional Transport in Magnetized Plasmas, Cambridge University Press (2002). The standard textbook derivation of the drift-kinetic equation, the linearized Fokker–Planck collision operator, and the neoclassical flux/flow moments that underpin Physics reference: the radially local drift-kinetic model.
P. Helander, Theory of plasma confinement in non-axisymmetric magnetic fields, Rep. Prog. Phys. 77, 087001 (2014). Review of stellarator neoclassical theory, ambipolarity, and the \(1/\nu\), \(\sqrt{\nu}\), and plateau regimes.
A. H. Boozer, Guiding center drift equations.
H. Sugama and S. Nishimura, How to calculate the neoclassical viscosity, diffusion, and current coefficients in general toroidal plasmas, Phys. Plasmas 9, 4637 (2002). Trajectory and moment-equation conventions relevant to the SFINCS full and DKES trajectory models.

SFINCS model, collision operator, and speed grid

M. Landreman, H. M. Smith, A. Mollén, and P. Helander, Comparison of particle trajectories and collision operators for collisional transport in nonaxisymmetric plasmas, Phys. Plasmas 21, 042503 (2014). The SFINCS paper: the radially local drift-kinetic model, the Delta/alpha/nu_n normalization, and the full-vs-DKES trajectory comparison implemented in dkx.drift_kinetic.
M. Mollén et al., Implementation of a full linearized Fokker-Planck collision operator in SFINCS. Basis for the Fokker–Planck operator in dkx.collisions.
M. Landreman and D. R. Ernst, New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions, J. Comput. Phys. 243, 130 (2013). The non-classical orthogonal-polynomial speed grid and Rosenbluth-potential field-term treatment implemented in dkx.phase_space.make_speed_grid() and the Rosenbluth-potential terms in dkx.collisions.
A. Redl et al., A new set of analytical formulae for the computation of the bootstrap current and the neoclassical conductivity in tokamaks, Phys. Plasmas 28, 022502 (2021). Bootstrap-current formula used as an analytic cross-check for \(\langle \mathbf{j}\cdot\mathbf{B}\rangle\).
O. Sauter, C. Angioni, and Y. R. Lin-Liu, Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime.
M. Landreman and E. J. Paul, Magnetic Fields with Precise Quasisymmetry for Plasma Confinement.

Block-tridiagonal Legendre solver and variational bounds

The tier-1 structured solve (Numerics and algorithms) eliminates the Legendre chain of the monoenergetic drift-kinetic equation with a block-tridiagonal factorization and a truncated-storage back-substitution; the variational transport-coefficient bounds (Reduced-model capabilities) bracket the monoenergetic \(D_{11}\) from the same discrete operator:

S. P. Hirshman, K. C. Shaing, W. I. van Rij, C. O. Beasley Jr., and E. C. Crume Jr., Plasma transport coefficients for nonsymmetric toroidal confinement systems, Phys. Fluids 29, 2951 (1986). Establishes the block-tridiagonal structure of the Legendre-mode monoenergetic drift-kinetic equation and the variational principle whose upper and lower bounds on the transport coefficients converge toward the true value from opposite sides as the Legendre resolution grows — the basis for dkx.variational.
F. J. Escoto, Fast and accurate calculation of the bootstrap current and radial neoclassical transport in low collisionality stellarator plasmas, PhD thesis (2025). Derives the tridiagonal structure of the Legendre-mode representation of the monoenergetic drift-kinetic equation and the block elimination that dkx.drift_kinetic.KineticOperator.to_block_tridiagonal() and dkx.solve.solve() exploit.

Geometry and benchmark configurations

C. D. Beidler et al., Benchmarking of the mono-energetic transport coefficients – results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS), Nucl. Fusion 51, 076001 (2011). Source of the analytic W7-X / LHD harmonic tables used by dkx.magnetic_geometry.FluxSurfaceGeometry.from_scheme() (geometry schemes 2/3/4) and of the monoenergetic benchmark coefficients.

Experimental and cross-code validation anchors

N. A. Pablant et al., Core radial electric field and transport in Wendelstein 7-X plasmas.
N. A. Pablant et al., Investigation of the ion-root solution in Wendelstein 7-X.
C. D. Beidler et al., Demonstration of reduced neoclassical energy transport in Wendelstein 7-X.

Upstream SFINCS technical notes and manuals

The long-form SFINCS technical documentation (the version-3 technical notes, the Fokker–Planck implementation note, the \(\Phi_1\) notes, and the SFINCS user manual) consists of unpublished upstream project documents. Following the repository policy of citing rather than vendoring, they are not redistributed here; their archival home is the upstream SFINCS project repository:

SFINCS project (github.com/landreman/sfincs)

The peer-reviewed reference for the model is the 2014 Physics of Plasmas paper cited above, and the physics/numerics that dkx relies on are reproduced in Theory from the upstream SFINCS notes, Drift-kinetic equation and system of equations, and Physics reference: the radially local drift-kinetic model.

JAX and differentiable programming

For implicit differentiation through linear solves (and other solver-aware workflows), see:

Testing, validation, and coverage methodology

The testing strategy is informed by scientific-software verification work and by empirical evidence that line coverage is useful for finding untested code but is not sufficient as a quality target:

U. Kanewala and J. M. Bieman, Testing Scientific Software: A Systematic Literature Review.
S. Segura et al., Metamorphic Testing: Testing the Untestable.
L. Inozemtseva and R. Holmes, Coverage Is Not Strongly Correlated with Test Suite Effectiveness.
Hypothesis property-based testing documentation.
Chex documentation for JAX test variants and assertions.
JAX checkify documentation for functionalized runtime checks.

Linear algebra and preconditioning

The solver stack in dkx draws on standard Krylov and preconditioning references:

Y. Saad and M. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 7(3), 1986.
H. A. van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG,” SIAM J. Sci. Stat. Comput. 13(2), 1992.
P. Sonneveld and M. B. van Gijzen, “IDR(s): A family of simple and fast algorithms for solving large nonsymmetric systems of linear equations,” SIAM J. Sci. Comput. 31(2), 2008.
M. A. Woodbury, “Inverting modified matrices,” Statistical Research Group Memo Report, 1950 (Woodbury identity / low‑rank updates).
G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins Univ. Press, 2013 (Schur complements, block factorization).
M. de Sturler, “Truncation strategies for optimal Krylov subspace methods,” SIAM J. Numer. Anal. 36(3), 1999 (GCRO/deflation concepts).
M. Benzi, “Preconditioning techniques for large linear systems: a survey,” J. Comput. Phys. 182(2), 2002.

Optimization-focused neoclassical workflows

Related reduced-model thesis and paper materials are useful for:

adjoint properties of drift-kinetic equations,
derivative-aware workflows for optimization,
and convergence/scaling studies that inform regression tests and benchmarks.

Recent applications (examples to prioritize)

The following papers motivate transport and optimization-oriented examples: