Analytical¶

Background¶

Analytical

Educational script demonstrating analytical concepts.

Code¶

```python """ Analytical

Educational script demonstrating analytical concepts. """

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

heat_equation_2d/analytical.py¶

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

import numpy as np from typing import Callable

def solve_analytical_2d(X: np.ndarray, Y: np.ndarray, t: float, initial_func: Callable, D: float, Lx: float, Ly: float, method: str = "eigenfunction", N: int = 20) -> np.ndarray: """ Basic analytical solution for 2D heat equation (simplified version).

For now, this provides a simple eigenfunction expansion for sine wave initial conditions.
"""
if method == "eigenfunction":
    # Only works for sine wave initial conditions
    try:
        # Try to detect if this is a sine wave
        # This is a simplified implementation
        return _solve_sine_eigenfunction(X, Y, t, D, Lx, Ly, N)
    except:
        # Fallback: return zeros if analytical solution fails
        print("Analytical solution not available for this initial condition")
        return np.zeros_like(X)
else:
    print(f"Analytical method '{method}' not implemented yet")
    return np.zeros_like(X)

def _solve_sine_eigenfunction(X: np.ndarray, Y: np.ndarray, t: float, D: float, Lx: float, Ly: float, N: int = 20) -> np.ndarray: """Simple eigenfunction solution for (1,1) sine mode.""" # For u(x,y,0) = sin(πx/Lx)sin(πy/Ly), the analytical solution is: # u(x,y,t) = sin(πx/Lx)sin(πy/Ly) * exp(-Dπ²(1/Lx² + 1/Ly²)*t)

lambda_11 = np.pi**2 * (1/Lx**2 + 1/Ly**2)
return np.sin(np.pi * X / Lx) * np.sin(np.pi * Y / Ly) * np.exp(-D * lambda_11 * t)

def validate_against_analytical_2d(numerical_solution: np.ndarray, X: np.ndarray, Y: np.ndarray, t: float, initial_func: Callable, D: float, Lx: float, Ly: float, method: str = "eigenfunction") -> dict: """ Basic validation against analytical solution. """ try: analytical_solution = solve_analytical_2d(X, Y, t, initial_func, D, Lx, Ly, method)

    # Calculate error metrics
    absolute_error = np.abs(numerical_solution - analytical_solution)
    relative_error = absolute_error / (np.abs(analytical_solution) + 1e-12)

    # Calculate grid spacing for integration
    dx = X[0, 1] - X[0, 0] if X.shape[1] > 1 else 1.0
    dy = Y[1, 0] - Y[0, 0] if Y.shape[0] > 1 else 1.0

    return {
        "max_abs_error": np.max(absolute_error),
        "max_rel_error": np.max(relative_error),
        "mean_abs_error": np.mean(absolute_error),
        "mean_rel_error": np.mean(relative_error),
        "l2_error": np.sqrt(dx * dy * np.sum(absolute_error**2)),
        "l2_norm_analytical": np.sqrt(dx * dy * np.sum(analytical_solution**2)),
        "relative_l2_error": np.sqrt(dx * dy * np.sum(absolute_error**2)) / 
                           (np.sqrt(dx * dy * np.sum(analytical_solution**2)) + 1e-12)
    }
except Exception as e:
    print(f"Validation failed: {e}")
    return {
        "max_abs_error": float('nan'),
        "max_rel_error": float('nan'), 
        "mean_abs_error": float('nan'),
        "mean_rel_error": float('nan'),
        "l2_error": float('nan'),
        "l2_norm_analytical": float('nan'),
        "relative_l2_error": float('nan')
    }

Placeholder functions to match the expected interface¶

def solve_analytical(x: np.ndarray, t: float, initial_func: Callable, D: float, L: float, method: str = "eigenfunction") -> np.ndarray: """1D analytical solution placeholder.""" # This is for 1D compatibility - just return zeros for now return np.zeros_like(x)

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

Exercises¶

Exercise 1. For the 2D heat equation with Dirichlet boundary conditions on \([0, L_x] \times [0, L_y]\), write the eigenfunction corresponding to mode numbers \((m, n)\) and its eigenvalue.

Exercise 2. Compute \(\lambda_{1,1}\) for a square domain \(L_x = L_y = 1\) with \(D = 0.01\). After what time \(t^*\) has this mode decayed to 1% of its initial amplitude?

Exercise 3. Why is the analytical solution for the 2D heat equation limited to certain initial conditions (e.g., sine waves)? What alternative approaches exist for arbitrary initial data?

Exercise 4. Define the relative \(L^2\) error between a numerical solution \(u_h\) and an analytical solution \(u_a\) on a 2D grid. Write the discrete approximation using grid spacings \(\Delta x\) and \(\Delta y\).