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warp-solver-equations\n\n**Generates** the Runge–Kutta 4 time‐integration update formulas for each evolved field \\(X\\) in a warp-bubble simulation, by parsing finite-difference stencil definitions from the [warp-discretization](/arcticoder/warp-discretization) repository and rendering them as LaTeX.\n\n## Features\n\n- **Stencil parsing**:  \n  Reads `stencil_*.tex` files, extracts grid‐point coefficients and offsets.\n- **Evolution definitions**:  \n  Imports metric components \\(g_{ij}\\) (and any auxiliary fields) from a template/symbolic definition.\n- **Discrete RHS construction**:  \n  Substitutes parsed stencil formulas into continuum equations \\(\\partial_t X = F(X,\\partial_r X,\\partial_\\theta X)\\) to build a Python function `F(X)`.\n- **RK4 embedding**:  \n  Symbolically defines and expands the four RK4 stages:\n```python\n  k1 = F(X_n)\n  k2 = F(X_n + dt/2 * k1)\n  k3 = F(X_n + dt/2 * k2)\n  k4 = F(X_n + dt   * k3)\n  X_{n+1} = X_n + dt/6 * (k1 + 2*k2 + 2*k3 + k4)\n```\n\n-   **LaTeX output**:  \n    Renders the $k_1\\!\\dots\\!k_4$ definitions and final $X^{n+1}$ formula for each field into a standalone `solver_update.tex`.\n    \n\n## Dependencies\n\n-   Python 3.7+\n    \n-   (no external Python libraries required beyond the standard library)\n    \n\n## Inputs\n\nCopy or symlink all `.tex` files from https://github.com/arcticoder/warp-discretization repo into an `inputs/` directory:\n\n-   `discretization.tex`\n    \n-   `stencil_r_2nd_order_001.tex`\n    \n-   `stencil_r_2nd_order_005.tex`\n    \n-   `stencil_r_4th_order_002.tex`\n    \n-   `stencil_r_4th_order_006.tex`\n    \n-   `stencil_theta_2nd_order_003.tex`\n    \n-   `stencil_theta_4th_order_004.tex`\n    \n\n## Usage\n\n```bash\ngit clone https://github.com/arcticoder/warp-solver-equations.git\ncd warp-solver-equations\n\n# Place or link the stencil .tex files into ./inputs\npython scripts/generate_solver_equations.py --input-dir inputs --output solver_update.tex\n```\n\n-   **`--input-dir`**: directory containing all `*.tex` stencil files.\n    \n-   **`--output`**: path to the generated LaTeX document (`solver_update.tex`).\n    \n\n## Output\n\n-   **`solver_update.tex`**: a self-contained LaTeX file showing, for each evolved variable $X$:\n    \n```math\n    \\[\n      k_1^X = F\\bigl(X^n,\\partial_r X^n,\\partial_\\theta X^n\\bigr)\n    \\]\n    \\[\n      k_2^X = F\\bigl(X^n + \\tfrac{\\Delta t}{2}k_1,\\dots\\bigr)\n    \\]\n    % … k_3, k_4 …\n    \\[\n      X^{n+1} = X^n + \\tfrac{\\Delta t}{6}\\bigl(k_1 + 2k_2 + 2k_3 + k_4\\bigr)\n    \\]\n```\n    \n\n## Dependence\n\n-   This repo depends on [warp-discretization](https://github.com/arcticoder/warp-discretization) for the stencil source files.\n    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