Grid¶

Background¶

Grid

Educational script demonstrating grid concepts.

Code¶

```python """ Grid

Educational script demonstrating grid concepts. """

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

heat_equation_1d/grid.py¶

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

import numpy as np from typing import NamedTuple

class GridParams(NamedTuple): """Grid parameters for the 1D heat equation.""" L: float # Length of domain T: float # Total time Nx: int # Number of spatial points Nt: int # Number of time steps D: float # Thermal diffusivity dx: float # Spatial step size dt: float # Time step size x: np.ndarray # Spatial grid coeff: float # Diffusion coefficient (D*dt/dx^2)

def create_grid(L: float = 1.0, T: float = 0.1, Nx: int = 100, Nt: int = 1000, D: float = 0.01) -> GridParams: """ Create spatial and temporal grids for the heat equation.

Args:
    L: Length of the rod
    T: Total simulation time
    Nx: Number of spatial grid points
    Nt: Number of time steps
    D: Thermal diffusivity constant

Returns:
    GridParams containing all grid information
"""
if L <= 0 or T <= 0 or D <= 0:
    raise ValueError("L, T, and D must be positive")
if Nx < 3 or Nt < 1:
    raise ValueError("Nx must be >= 3, Nt must be >= 1")

dx = L / (Nx - 1)
dt = T / Nt
x = np.linspace(0, L, Nx)
coeff = D * dt / dx**2

return GridParams(L, T, Nx, Nt, D, dx, dt, x, coeff)

def check_stability(coeff: float, method: str = "forward") -> None: """ Check stability condition for explicit schemes.

Args:
    coeff: Diffusion coefficient (D*dt/dx^2)
    method: Numerical method being used

Raises:
    ValueError: If stability condition is violated
"""
if method == "forward" and coeff > 0.5:
    raise ValueError(
        f"Forward Euler unstable: coeff = {coeff:.4f} > 0.5. "
        f"Reduce dt or increase dx."
    )

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

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

Exercises¶

Exercise 1. For a rod of length \(L = 1\) with \(N_x = 101\) grid points and thermal diffusivity \(D = 0.01\), compute the spatial step \(\Delta x\). If the time step is \(\Delta t = 10^{-4}\), compute the stability parameter \(\alpha = D\Delta t / \Delta x^2\) and determine whether Forward Euler is stable.

Exercise 2. Derive the stability condition \(\alpha \le 1/2\) for the Forward Euler discretization of the 1D heat equation using von Neumann analysis.

Exercise 3. If a simulation requires \(N_t = 10{,}000\) time steps with \(\Delta t = T/N_t\), and \(T = 0.1\), what is \(\Delta t\)? How would you adjust \(N_t\) if \(\Delta x\) is halved (and \(D\) remains unchanged) to maintain stability of Forward Euler?

Exercise 4. The GridParams named tuple stores the coefficient \(\alpha = D\Delta t / \Delta x^2\). Explain why this single parameter fully characterizes the behavior of the Forward Euler scheme for the heat equation (assuming Dirichlet boundary conditions and a fixed grid).