Add 31 PRBench task specifications (instructions, task.yaml, paper markdown; ground truth withheld)
a48f2b1 verified 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.
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) |
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:
- Methodology: Describe the three electromagnetic wave solver models (Truncated Taylor, Truncated Polynomial Fit, All-Orders)
- 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
- Algorithm Details: Describe the polynomial fitting procedure, finite difference stencil construction, and boundary condition implementation
- 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.
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 ($n_H = 0.05 n_e$, $n_D = 0.95 n_e$).
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).
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:
- 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.
- Extrapolate: Calculate predicted times for:
- $N_{2D} = 200^2 = 40,000$
- $N_{3D} = 200^3 = 8,000,000$
- 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:
- Exact: $I_0(0.5 k^2) e^{-0.5 k^2}$.
- Taylor: 4th Order Taylor expansion around $k=0$.
- Polynomial Fit: 4th Order fit over window $W=2$ (i.e., fit range $[-2, 2]$).
6. Code Structure
6.1 Core Modules
Path: reproduction/
plasma.py:- Functions for $n(x)$, $T(x)$, $B(x)$.
get_plasma_parameters(x)returning $\omega_{pj}, \Omega_{cj}, v_{th}, \rho_L$.
dielectric.py:calc_K_tensor(x, k_x, ...): Implements Eq (11)-(18).- Uses
scipy.special.wofzfor $Z(\zeta)$.
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.
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/
7. Deliverables Summary
Upon completion, the following files must exist:
reproduction/ANALYSIS.md— Your methodology and formula analysisreproduction/plasma.pythroughreproduction/run_sim.py— Core simulation modulesreproduction/fig2_profiles.pythroughreproduction/table1_scaling.py— Figure/table scriptsdata/fig2_profiles.csvthroughdata/table1_extrapolation.csv— Output data files (9 total)
Export data with 8 decimal places (%.8e).
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.