Grid¶

Background¶

Grid

Educational script demonstrating grid concepts.

Code¶

```python """ Grid

Educational script demonstrating grid concepts. """

============================================================================¶

heat_equation_2d/grid.py¶

============================================================================¶

import numpy as np from typing import NamedTuple

class Grid2DParams(NamedTuple): """Grid parameters for the 2D heat equation.""" Lx: float; Ly: float; T: float; Nx: int; Ny: int; Nt: int; D: float dx: float; dy: float; dt: float; x: np.ndarray; y: np.ndarray X: np.ndarray; Y: np.ndarray; rx: float; ry: float

def check_2d_stability(rx: float, ry: float, method: str = "forward") -> None: """ Check stability condition for 2D explicit schemes.

Args:
    rx: Diffusion coefficient in x direction (D*dt/dx^2)
    ry: Diffusion coefficient in y direction (D*dt/dy^2)
    method: Numerical method being used

Raises:
    ValueError: If stability condition is violated
"""
if method == "forward" and rx + ry > 0.5:
    raise ValueError(
        f"Forward Euler unstable in 2D: rx + ry = {rx + ry:.4f} > 0.5. "
        f"Reduce dt or increase dx/dy."
    )

def create_time_array_2d(T: float, Nt: int) -> np.ndarray: """Create time array for the 2D simulation.""" return np.linspace(0, T, Nt + 1)

def get_stability_info(params): """Get stability information for the grid parameters.""" stability_param = params.rx + params.ry return { "stability_parameter": stability_param, "is_stable_forward": stability_param <= 0.5, "rx": params.rx, "ry": params.ry, "dx": params.dx, "dy": params.dy, "dt": params.dt }

def create_2d_grid(Lx=1.0, Ly=1.0, T=0.1, Nx=50, Ny=50, Nt=1000, D=0.01): """Create 2D grid parameters.""" dx, dy, dt = Lx/(Nx-1), Ly/(Ny-1), T/Nt x, y = np.linspace(0, Lx, Nx), np.linspace(0, Ly, Ny) X, Y = np.meshgrid(x, y, indexing="ij") rx, ry = Ddt/dx2, Ddt/dy**2 return Grid2DParams(Lx, Ly, T, Nx, Ny, Nt, D, dx, dy, dt, x, y, X, Y, rx, ry)

if name == "main": pass ```

Exercises¶

Exercise 1. For a 2D grid with \(L_x = L_y = 1\), \(N_x = N_y = 50\), \(N_t = 1000\), \(T = 0.1\), and \(D = 0.01\), compute the stability parameters \(r_x\) and \(r_y\) and determine if Forward Euler is stable.

Exercise 2. Derive the 2D stability condition \(r_x + r_y \le 1/2\) for Forward Euler using von Neumann analysis with a 2D Fourier mode \(u_{j,k}^n = g^n e^{i(j\theta_x + k\theta_y)}\).

Exercise 3. If the 2D grid uses \(N_x = 100\) and \(N_y = 100\), how many unknowns are in the flattened system? Why do 2D implicit methods typically use sparse solvers?

Exercise 4. Explain the difference between indexing="ij" and indexing="xy" in np.meshgrid. Which convention does the 2D heat equation code use and why?