ALADIN Z-Pinch Benchmark v1.0

ALADIN Z-Pinch Benchmark v1.0 Cylindrical HLLD Kernels, Numerical Diagnostics and Full Verification Suite Overview ALADIN Z-Pinch Benchmark v1.0 is an initial research-software release within the ALADIN Plasma Drive project. It establishes a computational baseline for cylindrical magnetohydrodynamic (MHD) Z-pinch simulations using directional HLLD numerical flux kernels. The release provides three directional HLLD kernel implementations (X, Y and Z), a cylindrical residual operator, CFL timestep control, numerical diagnostics, and an integrated verification suite. The primary objective is to establish a reproducible computational foundation for investigating magnetized plasma dynamics, current-density distributions, Lorentz forces, radial plasma behaviour and numerical stability in cylindrical geometry. Software Components HLLD X kernel: Directional MHD numerical flux calculations. HLLD Y kernel: Directional MHD numerical flux calculations. HLLD Z kernel: Directional MHD numerical flux calculations. Cylindrical residual operator: Radial momentum and cylindrical geometric residual calculations. Z-pinch benchmark: Integrated simulation configuration with timestep control and diagnostics. Full verification suite: Twelve verification and diagnostic sections examining the numerical implementation and its behaviour. Benchmark archive: Packaged benchmark software and associated test materials. Full Verification Suite The v1.0 release includes twelve verification and diagnostic sections designed to examine different aspects of the Z-pinch implementation. 1. Core Z-Pinch Simulation Demonstrates the integrated execution of the directional HLLD kernels, CFL timestep calculation and plasma diagnostics in a cylindrical Z-pinch configuration. Examines: Density and pressure evolution. Radial and azimuthal velocity. Azimuthal magnetic field. Axial current density. Radial Lorentz force. Radial momentum residual. 2. Density-Floor Sensitivity Examines the sensitivity of the calculated core Lorentz force and radial velocity to changes in the numerical density floor. Demonstrates whether the reported core force remains consistent under the tested floor configurations. 3. Reverse-Current Test Reverses the signs of the azimuthal magnetic field and axial current density. Demonstrates the expected sign behaviour of the radial Lorentz force and verifies that reversing both quantities preserves the inward force direction in the tested configuration. 4. Independent Current-Density Reconstruction Compares the current density obtained from the simulation with an independently reconstructed current using the magnetic-field gradient. Demonstrates numerical agreement between the two current-density calculations away from the axis. 5. Axis Regularity Examines current-density behaviour near the cylindrical axis. Compares the analytical near-axis current with discrete numerical estimates to investigate the regularity of the cylindrical formulation. 6. Independent Lorentz-Force Verification Reconstructs the radial Lorentz force independently from the magnetic-field and current-density quantities. Compares the independently calculated force against the force reported by the simulation. Demonstrates strong numerical agreement across the sampled radial positions. 7. Momentum Closure Compares the measured radial momentum evolution against a separately assembled radial force budget. Evaluates the agreement between momentum evolution and the included physical force contributions. The initial results identify discrepancies that require further investigation. 8. Disk + Z-Pinch Configuration Extends the benchmark configuration to include a rotating disk alongside the Z-pinch magnetic structure. Examines: Density distribution. Radial and azimuthal velocities. Centrifugal contribution. Radial Lorentz force. Pressure-gradient contribution. This test provides an initial integrated disk-plus-pinch experiment rather than a completed equilibrium validation. 9. CFL Decomposition Examines the timestep controller under evolving plasma conditions. Tracks: Local timestep. Radial position of the limiting cell. Density. Radial velocity. Characteristic fast-wave speed. Demonstrates the timestep response to changes in local characteristic speeds, including timestep reduction as the simulated state becomes increasingly demanding. 10. Outer-Disk Inspection Examines the radial evolution of density, velocity, azimuthal magnetic field and axial current density near the outer computational boundary. Provides diagnostic information about the outer-disk state and identifies regions requiring additional stability and boundary-condition investigation. 11. Full Momentum Closure Compares the measured radial momentum update against the discrete momentum flux-divergence contribution used by the numerical update. Demonstrates exact reported agreement in the tested discrete update comparison. This is an implementation consistency check and does not independently establish full physical momentum-budget conservation. 12. Extreme-Cell Instrumentation Tracks the most extreme computational cells during the early evolution of the Z-pinch simulation. Records: Radial velocity. Density. Conserved momentum variables. Total energy. Azimuthal magnetic field. Numerical fluxes at selected faces. Provides diagnostic visibility into the development of extreme states and helps identify potential sources of numerical instability. Verification Summary The verification campaign provides evidence for several important numerical properties: Independent current-density reconstruction shows close agreement with the simulation-derived current away from the axis. Independent Lorentz-force reconstruction shows close agreement with the calculated radial force. Reverse-current behaviour produces the expected inward-force direction when both current and magnetic field are reversed. Near-axis current-density behaviour is examined against an analytical reference. Density-floor sensitivity is investigated across multiple configurations. Discrete momentum-update consistency is examined. CFL timestep behaviour is monitored under evolving plasma conditions. Extreme-cell instrumentation provides detailed information about potentially unstable regions. The initial verification results also identify areas requiring additional work, including independent physical momentum-budget closure, positivity preservation, long-term stability, conservation properties and resolution convergence. These limitations are retained as part of the documented v1.0 baseline. Scope and Development Status ALADIN Z-Pinch Benchmark v1.0 is an initial numerical software baseline, not a claim of completed physical validation or a universally validated MHD benchmark. Its purpose is to establish a reproducible reference implementation and verification record for subsequent development. Future work will focus on: Improving the directional HLLD numerical operators. Strengthening conservation and stability properties. Extending physical and numerical validation. Conducting systematic resolution and timestep convergence studies. Integrating additional plasma physics operators. Incorporating the Z-pinch solver into the broader ALADIN Plasma Drive framework. Release Information Project: ALADIN Plasma Drive Software: ALADIN Z-Pinch Benchmark Version: 1.0 Release type: Initial numerical baseline Domain: Computational plasma physics and magnetohydrodynamics Geometry: Cylindrical Numerical method: Directional HLLD-based flux calculations Verification: Twelve verification and diagnostic sections ALADIN Z-Pinch Benchmark v1.0 establishes the initial computational and verification foundation for the continued development of the ALADIN Plasma Drive framework.