# Paper Reproduction Task

You are given a plasma physics paper to reproduce. Your goal is to read the paper, understand its methodology, and implement electromagnetic wave simulations to reproduce 7 key figures and 1 table.

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## 1. Article Information

| Field | Value |
|-------|-------|
| **Title** | Accelerating simulations of electromagnetic waves in hot, magnetized fusion plasmas |
| **Authors** | R H S Budé, D Van Eester, J van Dijk, R J E Jaspers and A B Smolders |
| **DOI** | 10.1088/1361-6587/abd619 |
| **Paper File** | `Bude2021.md` (in the same task directory) |

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## 2. What You Must Do

### Step 1: Read and Analyze the Paper
Read the full paper at `Bude2021.md`. Write a detailed analysis document at `reproduction/ANALYSIS.md` that covers:

1. **Methodology**: Describe the three electromagnetic wave solver models (Truncated Taylor, Truncated Polynomial Fit, All-Orders)
2. **Key Formulas**: List and explain every formula needed for the reproduction, including:
   - The 1D Helmholtz equation
   - Hot plasma dielectric tensor (Stix/Swanson notation)
   - Plasma Dispersion Function
   - Differential operator replacement for polynomial fit model
   - RRSE error metric
   - Power absorption calculation
3. **Algorithm Details**: Describe the polynomial fitting procedure, finite difference stencil construction, and boundary condition implementation
4. **Physical Observables**: Explain what each figure/table measures and its physical significance

### Step 2: Implement the Simulation
Write all code under `reproduction/`. You need:
- Core simulation modules (see Section 6 below)
- Individual scripts for each figure/table

### Step 3: Generate Data Files
Run your scripts to produce CSV data files under `data/`. See Section 5 for exact specifications.

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## 3. Input Parameters

### 3.1 Machine Parameters (JET-like)
| Parameter | Symbol | Value | Unit |
|-----------|--------|-------|------|
| Major Radius | $R_0$ | 2.97 | m |
| Minor Radius | $a$ | 1.05 | m |
| Spill-over Radius | $a_s$ | 1.35 | m |
| Magnetic Field Axis | $B_0$ | 3.45 | T |
| Toroidal Mode | $n_{\text{tor}}$ | 27 | - |
| Frequency | $f$ | 51.0 | MHz |

### 3.2 Plasma Profiles
**Density ($n$) and Temperature ($T$):**
- **Core ($|x| < a_p$)**: $G_{\text{core}}(x) = (G_0 - G_p)(1 - (x/a_p)^2)^\alpha + G_p$
- **Edge ($|x| \ge a_p$)**: $G_{\text{edge}}(x) = G_p e^{-|x-a_p|/\lambda}$
- Transition point $a_p = 0.95$ m.

**Specific Coefficients:**
| Profile | Axis Value ($G_0$) | Edge Value ($G_s$) | $\alpha$ | $\lambda$ |
|---------|--------------------|--------------------|----------|-----------|
| Density ($n_e, n_i$) | $7 \times 10^{19}$ m$^{-3}$ | $2 \times 10^{19}$ m$^{-3}$* | 1.0 | 0.05 |
| Temperature ($T_e, T_i$) | 5 keV | 100 eV | 1.5 | 0.05 |

*Note: Paper text says $n_s = 2 \times 10^{19}$, $G_p$ must be calculated to ensure continuity or assumed $G_p \approx G_s$ given the context.*

**Species:**
- Deuterium (Major): Fully ionized.
- Hydrogen (Minor): 5% concentration ($n_H = 0.05 n_e$, $n_D = 0.95 n_e$).

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## 4. Banned Libraries & Practices

The following are **NOT allowed**:

| Banned | Reason |
|--------|--------|
| `plasma-py` (simulation modules) | Must implement dielectric tensor and wave solvers from scratch. |
| Closed-source solvers | Code must be pure Python/NumPy/SciPy. |
| Symbolic Math (SymPy) | Numerical differentiation/fitting should be used as per the paper. |

**Allowed:** `numpy`, `scipy` (specifically `scipy.special` for Bessel/Error functions, `scipy.sparse`, `scipy.linalg`).

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## 5. Data Files to Reproduce

### 5.1 Figure 2: Plasma Profiles
**File**: `data/fig2_profiles.csv`
- **X-axis**: $x$ [m] (-1.5 to 1.5)
- **Y-axes**: $n_e$ [$10^{19}$ m$^{-3}$], $T_e$ [keV]
- **Description**: Verify the parabolic core and exponential edge implementation.

### 5.2 Figures 3 & 4: Dielectric Tensor Approximation
**File**: `data/fig3_4_tensor_fit.csv`
- **Context**: $x = -0.1$ m (Hydrogen resonance layer).
- **X-axis**: Normalized wavenumber $k_x \rho_{LH}$ (-3 to 3).
- **Y-axes**: $\text{Re}(\epsilon_{11})$.
- **Series**: Calculated exact values, 2nd Order Taylor, 6th Order Polynomial Fit.
- **Fit Window**: $W = 1.0$ (approx, check paper visual).

### 5.3 Figure 5: Electric Field Comparison
**File**: `data/fig5_efield.csv`
- **Domain**: $-0.9 < x < -0.1$ m (IBW region).
- **Resolution**: $N = 2017$.
- **Y-axis**: $|E_x|$ [V/m].
- **Series**: Truncated Taylor, Truncated Polynomial ($N_p=8, W=1.2$).
- **Note**: The paper compares these against All-Orders. If All-Orders is too expensive, comparing Taylor vs Poly is sufficient.



### 5.4 Figure 6: Absorbed Power Distribution
**File**: `data/fig6_power.csv`
- **Description**: Comparison of power absorption profiles derived from Poynting flux (Eq 50).
- **X-axis**: $x$ [m] (-1.5 to 1.5).
- **Y-axis**: $P_{abs}$ [arbitrary units].
- **Series**:
    - Per Species: Electron (e), Hydrogen (H), Deuterium (D).
    - Total Power.
- **Models**:
    - Truncated Taylor (illustrating unphysical oscillations/smearing).
    - Truncated Polynomial Fit ($N_p=8$) (proxy for All-Orders if necessary, as they are visually identical in the paper).
- **Calculation**: $\vec{J}_p$ must be calculated by applying the respective differential operator ($\bar{\bar{M}}_{op}$) to the solution field $\vec{E}$.

### 5.5 Figure 7: Convergence (RRSE)
**File**: `data/fig7_error_scaling.csv`
- **X-axis**: Number of grid points $N$ (Log scale, 100 to 10,000).
- **Y-axis**: RRSE (Log scale).
- **Series**: Truncated Taylor, Truncated Polynomial Fit ($N_p=8$).
- **Reference**: High resolution Polynomial Fit ($N=12601$) can act as proxy for "All-Orders" if full spectral code is not implemented.

### 5.6 Figure 8: Computational Time
**File**: `data/fig8_timing.csv`
- **X-axis**: $N$ (Log scale).
- **Y-axis**: Time [s] (Log scale).
- **Series**: Matrix Setup Time, Matrix Solve Time (for Poly and Taylor models).

### 5.7 Table 1: Computational Cost Extrapolation
**File**: `data/table1_extrapolation.csv`
- **Description**: Extrapolation of runtime scaling to 2D and 3D cases based on 1D benchmarks.
- **Inputs**: Slope ($\alpha$) and intercept data derived from Figure 8 (Timing).
- **Methodology**:
    1.  **Fit**: Perform linear regression on log-log timing data (from Fig 8) for $N > 2000$ to obtain scaling laws $T \propto N^\alpha$ for Setup and Solve phases.
    2.  **Extrapolate**: Calculate predicted times for:
        -   $N_{2D} = 200^2 = 40,000$
        -   $N_{3D} = 200^3 = 8,000,000$
    3.  **Dimensional Scaling**: Apply the paper's approximation that computational time scales with stencil size $s$ raised to the dimension $D$ ($t \propto s^D$).
        -   Truncated Taylor: $s=3$.
        -   Truncated Polynomial: $s=9$ (for $N_p=8$).
- **Output Columns**: Model Name, Matrix Setup $\alpha$, Matrix Invert $\alpha$, Stencil Size, Time (1D), Time (2D), Time (3D).

### 5.8 Figure 9: Parameter Sweep (RRSE vs Fit Window)
**File**: `data/fig9_param_sweep.csv`
-   **Context**: Section 6, analyzing the sensitivity of the Truncated Polynomial Fit model to the window size $W$ and polynomial order $N_p$.
-   **Fixed Parameter**: Grid resolution $N = 2017$.
-   **X-axis**: Fit Window $W$ ($0.5$ to $3.0$, step $\approx 0.1$).
-   **Y-axis**: RRSE (Log scale).
-   **Series**:
    -   Polynomial Orders: $N_p \in \{4, 6, 8, 10, 12\}$.
    -   Baseline: Truncated Taylor model error (constant line).
-   **Reference**: Use a high-resolution simulation ($N \approx 12601$, $N_p=12$) as the ground truth for error calculation.

### 5.9 Figure 10: Component Approximation
**File**: `data/fig10_function_approx.csv`
-   **Context**: Section 7 (Discussion), comparing approximation methods for the specific term $f(k) = I_n(\lambda)e^{-\lambda}$ (where $\lambda = \frac{1}{2} k^2 \rho^2$).
-   **Parameters**: Use $n=0$ (Modified Bessel function of the first kind) and normalized units where $\rho=1$.
-   **X-axis**: Normalized wavenumber $k \rho$ (Range: $-2.5$ to $2.5$).
-   **Y-axis**: Function Value.
-   **Series**:
    1.  **Exact**: $I_0(0.5 k^2) e^{-0.5 k^2}$.
    2.  **Taylor**: 4th Order Taylor expansion around $k=0$.
    3.  **Polynomial Fit**: 4th Order fit over window $W=2$ (i.e., fit range $[-2, 2]$).

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## 6. Code Structure

### 6.1 Core Modules
**Path**: `reproduction/`

1.  `plasma.py`:
    - Functions for $n(x)$, $T(x)$, $B(x)$.
    - `get_plasma_parameters(x)` returning $\omega_{pj}, \Omega_{cj}, v_{th}, \rho_L$.
2.  `dielectric.py`:
    - `calc_K_tensor(x, k_x, ...)`: Implements Eq (11)-(18).
    - Uses `scipy.special.wofz` for $Z(\zeta)$.
3.  `solvers.py`:
    - `TaylorSolver`: Constructs sparse matrix using Eq (40)-(43).
    - `PolyFitSolver`:
        - Performs polynomial fitting of tensor elements.
        - Constructs sparse matrix using Eq (46)-(48).
        - Handles Finite Difference stencils for high order derivatives.
4.  `run_sim.py`:
    - Orchestrates the grid generation, solving, and data export.

### 6.2 Figure Scripts
| Figure/Table | Script Path |
|--------|-------------|
| Fig 2 | `reproduction/fig2_profiles.py` |
| Fig 3,4 | `reproduction/fig3_4_tensor.py` |
| Fig 5 | `reproduction/fig5_fields.py` |
| Fig 6 | `reproduction/fig6_power.py` |
| Fig 7 | `reproduction/fig7_convergence.py` |
| Fig 8 | `reproduction/fig8_timing.py` |
| Fig 9 | `reproduction/fig9_params.py` |
| Fig 10 | `reproduction/fig10_approx.py` |
| Table 1 | `reproduction/table1_scaling.py` |

### 6.3 Analysis Document
**Path**: `reproduction/ANALYSIS.md`

Must contain your analysis of the paper's methodology and all formulas needed for reproduction.

### 6.4 Output Directories
- Data files: `data/`
- Code files: `reproduction/`

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## 7. Deliverables Summary

Upon completion, the following files must exist:
1. `reproduction/ANALYSIS.md` — Your methodology and formula analysis
2. `reproduction/plasma.py` through `reproduction/run_sim.py` — Core simulation modules
3. `reproduction/fig2_profiles.py` through `reproduction/table1_scaling.py` — Figure/table scripts
4. `data/fig2_profiles.csv` through `data/table1_extrapolation.csv` — Output data files (9 total)

Export data with 8 decimal places (`%.8e`).

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## 8. Reproduction Requirements

### 8.1 Computational Constraints

| Requirement | Value | Notes |
|-------------|-------|-------|
| **Memory** | 8 GB | High N simulations (10k+) require sparse matrices. |
| **Precision** | Complex Double | `complex128` in numpy is essential. |
| **Optimization** | Vectorization | Tensor calculation must be vectorized over $k$ or $x$. |
| **Total Time Limit** | ~4 hours | For all figures, assuming 1D models only. |

### 8.2 Output Format
-   **Data Files**: CSV format with headers.
-   **Precision**: 8 decimal places for floating point values.

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