{"id":28945301,"url":"https://github.com/arcticoder/warp-discretization","last_synced_at":"2026-04-29T10:07:57.595Z","repository":{"id":300406657,"uuid":"993208153","full_name":"arcticoder/warp-discretization","owner":"arcticoder","description":"Generates finite-difference spatial‐derivative stencils and Taylor-series error-order expansions from the closed-form warp-bubble expressions in final_expressions.tex, and outputs a ready-to-compile LaTeX file 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Discretization\n\n[![Python](https://img.shields.io/badge/Python-3.8+-blue.svg)](https://python.org)\n[![SymPy](https://img.shields.io/badge/SymPy-Latest-green.svg)](https://sympy.org)\n[![LaTeX](https://img.shields.io/badge/LaTeX-Required-red.svg)](https://latex-project.org)\n\n**Automated finite-difference stencil generation for warp bubble spacetime discretization**\n\nThis repository provides computational tools to generate finite-difference stencils and truncation-error expansions for spatial derivatives in warp-bubble spacetime expressions, enabling numerical simulation of exotic spacetime geometries.\n\n## Key Features\n\n- **Finite-Difference Stencils**: Automated generation of central difference formulas (2nd, 4th, 6th order)\n- **Error Analysis**: Complete Taylor-series truncation error characterization\n- **LaTeX Output**: Publication-ready discretization documentation\n- **SymPy Integration**: Symbolic mathematics for exact coefficient computation\n- **Pipeline Ready**: Integrates seamlessly with upstream warp-bubble repositories\n\n## Mathematical Framework\n\nThe tool processes closed-form warp-bubble expressions to generate numerical discretization schemes:\n\n### Input: Analytic Expressions\n- **Metric Components**: g_μν(x) from warp-bubble geometry\n- **Curvature Invariants**: R, R_μν R^μν, Riemann tensor components  \n- **Stress-Energy Tensor**: T_μν(x) exotic matter distributions\n- **Source**: `final_expressions.tex` from upstream pipeline\n\n### Output: Discretization Schemes\n- **Spatial Derivative Stencils**: Central finite-difference formulas for ∇_i, ∇_i∇_j operators\n- **Error Order Expansions**: Complete O(h²), O(h⁴), O(h⁶) truncation analysis\n- **Coefficient Tables**: Exact symbolic coefficients for each stencil point\n- **LaTeX Documentation**: Ready-to-compile discretization.tex with all formulas\n\n### Supported Orders\n- **2nd Order**: O(h²) accuracy, 3-point stencils\n- **4th Order**: O(h⁴) accuracy, 5-point stencils  \n- **6th Order**: O(h⁶) accuracy, 7-point stencils\n- **Custom**: User-configurable stencil orders and grid arrangements\n\n## Installation \u0026 Setup\n\n### Prerequisites\n```bash\n# Core dependencies\npip install sympy\u003e=1.9 numpy scipy matplotlib\n\n# LaTeX distribution (for output compilation)\n# Ubuntu/Debian: sudo apt-get install texlive-full\n# macOS: brew install mactex\n# Windows: Install MiKTeX or TeX Live\n```\n\n### Repository Setup\n```bash\ngit clone https://github.com/arcticoder/warp-discretization.git\ncd warp-discretization\npip install -r requirements.txt\n```\n\n## Usage Guide\n\n### Basic Stencil Generation\n```bash\n# Generate all discretization stencils\npython scripts/generate_stencils.py --input final_expressions.tex --output discretization.tex\n\n# Specify stencil order\npython scripts/generate_stencils.py --input final_expressions.tex --output discretization.tex --order 4\n\n# Generate with error analysis\npython scripts/generate_stencils.py --input final_expressions.tex --output discretization.tex --error-analysis\n```\n\n### Advanced Options\n```bash\n# Custom grid spacing\npython scripts/generate_stencils.py --input final_expressions.tex --spacing-symbol \"Delta_x\"\n\n# Export individual stencils\npython scripts/generate_stencils.py --input final_expressions.tex --export-individual --output-dir stencils/\n\n# Generate validation code\npython scripts/generate_stencils.py --input final_expressions.tex --generate-tests\n```\n\n## Output Structure\n\n```\ndiscretization.tex          # Main LaTeX document with all stencils\nstencils/                   # Individual stencil files (optional)\n├── first_derivatives.tex   # ∂/∂x, ∂/∂y, ∂/∂z stencils\n├── second_derivatives.tex  # ∂²/∂x², ∂²/∂y², ∂²/∂z² stencils\n├── mixed_derivatives.tex   # ∂²/∂x∂y, etc. stencils\n└── error_analysis.tex      # Truncation error bounds\n```\n\n## 🔗 Pipeline Integration\n\nThis repository is part of the comprehensive warp-bubble simulation pipeline:\n\n### Upstream Dependencies\n- **[warp-bubble-assemble-expressions](../warp-bubble-assemble-expressions)**: Provides `final_expressions.tex`\n- **[warp-bubble-einstein-equations](../warp-bubble-einstein-equations)**: Stress-energy tensor T_μν\n- **[warp-bubble-exotic-matter-density](../warp-bubble-exotic-matter-density)**: Energy density analysis\n\n### Downstream Applications  \n- **[warp-solver-equations](../warp-solver-equations)**: Time integration schemes\n- **[warp-solver-validation](../warp-solver-validation)**: Numerical validation framework\n- **Numerical relativity codes**: Grid-based spacetime evolution\n\n### Data Flow\n```\nAnalytic Expressions → Finite-Difference Stencils → Time Integration → Validation\n```\n\n## 🧪 Validation \u0026 Testing\n\n### Automated Tests\n```bash\n# Run all validation tests\npython -m pytest tests/ -v\n\n# Test stencil accuracy\npython tests/test_stencil_accuracy.py\n\n# Validate error bounds\npython tests/test_error_analysis.py\n```\n\n### Benchmarking\n```bash\n# Compare with analytical derivatives\npython scripts/benchmark_accuracy.py\n\n# Performance profiling\npython scripts/profile_generation.py\n```\n\n## 📖 Mathematical Details\n\n### Finite-Difference Formulas\nFor a scalar field f(x), the nth-order central difference approximation:\n\n```\nf'(x) ≈ (1/h) Σ c_i f(x + ih)    [2nd order: c = [-1/2, 0, 1/2]]\nf''(x) ≈ (1/h²) Σ c_i f(x + ih)   [2nd order: c = [1, -2, 1]]\n```\n\n### Error Analysis\nTruncation error bounds for each stencil:\n```\n|Error| ≤ (h^p/p!) max|f^(p)(ξ)|    where p = order + 1\n```\n\n### Grid Considerations\n- **Uniform spacing**: h = constant throughout domain\n- **Boundary handling**: One-sided and modified stencils near boundaries\n- **Stability analysis**: CFL conditions for time-stepping schemes\n\n## Applications\n\n- **Numerical Relativity**: Spacetime evolution simulations\n- **Warp Drive Research**: Exotic matter field dynamics\n- **Computational Physics**: General PDE discretization\n- **Algorithm Development**: Finite-difference method validation\n\n## Contributing\n\nContributions welcome! Areas of interest:\n- Higher-order stencil implementations\n- Adaptive grid refinement support\n- GPU acceleration for large-scale problems\n- Integration with finite element methods\n\n## References\n\n1. **Finite Difference Methods**: LeVeque, R. J., \"Finite Difference Methods for ODEs and PDEs\"\n2. **Numerical Relativity**: Baumgarte \u0026 Shapiro, \"Numerical Relativity: Solving Einstein's Equations on the Computer\"\n3. **Warp Drive Physics**: Alcubierre, M., Phys. Rev. D 53, 3571 (1994)\n\n---\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Fwarp-discretization","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Farcticoder%2Fwarp-discretization","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Farcticoder%2Fwarp-discretization/lists"}