[PyTokEq Enhancement] Improve q-profile accuracy to <5% error

[callme-YZ]

Problem

Current: q(axis) has ~30% error due to cylindrical approximation

From validation notes:

• Core safety factor不准确
• Limits physics fidelity
• May affect ballooning threshold analysis

Root Cause

Cylindrical approximation near magnetic axis:

• Current model uses simplified geometry
• Toroidal effects not fully captured
• Higher-order terms neglected

Proposed Solution

Option 1: Toroidal correction formula

• Implement Shafranov shift correction
• Use toroidal expansion (ε terms)
• Effort: Medium (1-2 weeks)

Option 2: Field-line integration

• Direct integration along field lines
• More accurate but slower
• Effort: High (3-4 weeks)

Option 3: Adaptive grid near axis

• Refined mesh at r→0
• Better resolution of core region
• Effort: Medium-high (2-3 weeks)

Deliverables

• Implement chosen method
• Test on Solovev (should achieve machine precision)
• Test on realistic equilibria (target <5% error)
• Update validation suite
• Document methodology

Success Criteria

• ✅ q(axis) error < 5% on standard benchmarks
• ✅ No performance degradation
• ✅ Validated on 3+ test cases

Priority

P2-medium 🟡 - Important for physics fidelity, not blocking

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Reported by: 小P ⚛️
Date: 2026-03-23