VEQ: a fast parametric Grad--Shafranov solver for fixed-boundary tokamak equilibria with flexible source profiles

Abstract: Veloce EQuilibrium (VEQ) is a compact parametric framework for tokamak modeling workflows that repeatedly query continuous fixed-boundary equilibria at low latency. The VEQPy implementation evaluated here is an axisymmetric fixed-boundary Grad-Shafranov solver whose main solve enforces a variationally induced projected residual. Its active unknowns are MXH-type flux-surface harmonics and shifted-Chebyshev coefficients for radial profile and source closures. Six input routes accept pressure-gradient, toroidal-field-function, poloidal-flux-gradient, enclosed toroidal current, current-density and safety-factor information through route-specific closures, while all routes map to the same finite-dimensional residual operator. Controlled tests show route consistency for smooth, mutually compatible inputs generated from a common reference equilibrium. For Pareto-selected reduced configurations in three G-EQDSK cases, the most accurate selected rows correspond to a D-shaped case (9 active parameters, minor-radius-normalized shape error 1.4e-3, solve-only median 1.6 ms), an H-mode case (65, 1.1e-3, 19 ms), and an X-point case treated as a smoothed fixed-boundary representation of a diverted boundary (94, 1.9e-3, 15 ms). Sampled pointwise strong-form Grad-Shafranov diagnostics show that enriching the active representation mainly improves interior force balance, whereas the global RMS and maximum values for the H-mode and X-point cases remain dominated by near-boundary contributions. In an isolated one-dimensional transport-geometry coupling test against the target geometry read from G-EQDSK, the temperature-profile response remains below about one percent. These results support using VEQ for repeated equilibrium-geometry queries, provided that pointwise diagnostics are retained to screen cases requiring boundary refinement, local correction or higher-fidelity equilibrium solves.

• The paper’s main contribution is presenting VEQ, a unified parametric Grad–Shafranov solver that supports six distinct input routes for fixed-boundary tokamak equilibrium calculations.
• It employs an MXH-Chebyshev parameterization and variational projected residuals to achieve high geometric fidelity with millisecond-scale solve latencies and RMS errors below 10⁻³.
• Benchmark results using analytic and G-EQDSK-derived equilibria validate VEQ’s performance, highlighting its suitability for real-time control and integrated simulation workflows.

Fast Parametric Grad–Shafranov Solves for Tokamak Equilibria: The VEQ Framework

Overview

The paper "VEQ: a fast parametric Grad–Shafranov solver for fixed-boundary tokamak equilibria with flexible source profiles" (2606.11821) introduces VEQ, a parametric equilibrium framework specifically designed for repeated high-latency queries of fixed-boundary axisymmetric MHD equilibria in tokamak scenario workflows. The VEQPy implementation leverages a finite-dimensional MXH-type flux-surface and shifted-Chebyshev radial profile parameterization, enabling rapid nonlinear Grad–Shafranov solves with variationally induced projected residuals. The solver supports six physically distinct input routes (pressure-gradient, toroidal-field-function, poloidal-flux-gradient, enclosed toroidal current, current-density, safety-factor), unifying them in a single residual assembly. Consistency and fidelity are rigorously benchmarked using both analytic and G-EQDSK-derived equilibria, emphasizing geometry diagnostics, strong-form force-balance residuals, and downstream impacts on transport-geometry channels.

Parametric Representation and Variational Residual Formulation

VEQ departs from traditional grid-based poloidal-flux field representations, employing instead MXH (Miller-type extended harmonics) angular basis for flux-surface geometry and shifted Chebyshev polynomials for radial dependence. The surface map, parameterized as $(R(\rho, \theta), Z(\rho, \theta))$, allows for continuous nested-surface equilibria that retain geometric richness with a controlled number of shape coefficients. Source and profile closures are similarly handled in the Chebyshev basis, supporting low- and high-order active families.

The central residual is constructed variationally from the Grad–Shafranov energy functional, resulting in a projected system where the test space is induced by shape variations in the ansatz. All input routes are mapped into a common canonical set of source profiles (pressure-gradient, toroidal-field source, etc.), ensuring uniformity in residual assembly. The key computational leg is a compact Petrov-Galerkin system, with shape-block equations enforcing orthogonality of the transformed Grad–Shafranov residual to admissible parameter-space variations.

Input Route Flexibility

To accommodate diverse workflow requirements and available data, VEQ supports six input routes (PF, PP, PI, PJ1, PJ2, PQ), each corresponding to different physical information: pressure gradients, current diagnostics, safety-factor profiles, etc. Route-specific profile closures convert these heterogeneous inputs to a consistent normalized source format. This route abstraction enables VEQ to seamlessly integrate with experimental data, analytic models, and control pipelines without modifying the core equilibrium formulation.

Numerical Benchmarks and Validation

VEQPy is extensively validated through controlled analytic benchmarks and realistic G-EQDSK-derived equilibria:

• Input Route Consistency: When provided with mutually compatible inputs, all six routes demonstrate convergence to identical flux-surface geometry and profile diagnostics, with discrepancies appearing only in near-edge quantities (e.g., $q_{95}$ sensitivity due to radial interpolation and boundary discretization).
• G-EQDSK Reconstruction: High-order VEQ configurations (i.e., with a large number of active coefficients) achieve robust RMS shape reconstruction errors, normalized to the minor radius, ranging from $7.3 \times 10^{-4}$ to $1.7 \times 10^{-3}$ for challenging Solov’ev, CHEASE, and EFIT test cases.
• Pareto Analysis: Systematic exploration of the accuracy-latency trade-off reveals that with $\mathcal{O}(10)$–$\mathcal{O}(100)$ parameters, VEQ can produce reduced-order models with RMS shape errors $<10^{-3}$ while maintaining millisecond-scale solve latency (e.g., 1.56 ms for 9 parameters in the D-shaped configuration, 19 ms for 65 parameters in H-mode).

Strong-Form Residual Diagnostics

A central claim is that VEQ enforces a projected residual system rather than pointwise satisfaction of the Grad–Shafranov equation. Diagnostics based on strong-form (cylindrical) residuals demonstrate that while the projected system effectively controls interior force-balance violations as the active order increases, strong-form violations localize near the plasma boundary in cases with complex edge geometry (notably in H-mode and X-point equilibria). This is intrinsic to both the fixed-boundary ansatz and the parametric closure, with boundary corrections strictly limited by the representation.

The implication is that for applications with stringent local force-balance or edge stability requirements, users must retain strong-form diagnostic screening and, if necessary, employ local corrections or couple to high-fidelity post-processors.

Downstream Modeling Impact

To characterize the impact of parametric equilibrium errors on workflow-relevant observables, the authors propagate geometry errors through a scope-controlled one-dimensional transport operator. For all G-EQDSK-based cases and reduced models tested, the induced temperature-profile and stored-energy errors stay below 1%, indicating that, for geometry-dominated channels, the VEQ representation is sufficient for most integrated modeling purposes.

Practical Implications

VEQ occupies a middle ground between analytic engineering closures (e.g., Miller-type) and fully grid-based equilibrium codes (e.g., EFIT, CHEASE, ECOM), combining geometry flexibility, low-latency repeated queries, and explicit diagnostic channels. The software design decouples setup and solve phases, leveraging precomputed spectral arrays and JIT-compiled kernels, thus achieving high efficiency.

The modularity and speed of VEQ make it natively compatible with real-time scenario exploration, control-oriented modeling, and rapid parameter scans, especially in systems where equilibrium geometry is repeatedly queried and incremental geometry correction is either unnecessary or tractable via diagnostic feedback.

Limitations and Future Work

The present implementation is strictly fixed-boundary and axisymmetric. Limitations include:

• No free-boundary capability or coupling to external equilibrium constraints (e.g., coils, passive structures, separatrix tracking)
• No direct treatment of 3D or stellarator equilibria
• Limited fidelity near critical points (X-point, separatrix, divertor regions), as illustrated by strong-form residual localization

Extensions toward free-boundary, 3D representations, and improved local boundary correction algorithms are prospective directions. The framework is also a candidate for warm-start initialization within high-fidelity solvers or advanced controlled scenario design.

Conclusion

VEQ provides a technically rigorous and efficient means for parametric fixed-boundary MHD equilibrium calculation in tokamak workflows. Its core innovation is the unification of flexible input routes, compact MXH-Chebyshev manifold parameterization, and variational projected residual enforcement, yielding high geometric fidelity at millisecond-scale solver latencies. Explicit diagnostic separation between projected and pointwise force-balance errors ensures transparency for stability analysis and scenario modeling. For applications not dominated by local edge physics, VEQ offers a compelling equilibrium backend for integrated simulation and control environments.