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Heat Equation 2D (Basic)

Background

Heat Equation 2D Basic

Educational script demonstrating heat equation 2d basic concepts.

Code

```python """ Heat Equation 2D Basic

Educational script demonstrating heat equation 2d basic concepts. """

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heat_equation_2d_BASIC.py

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import heat_equation_2d as he2

Example usage

def main(): """Main example demonstrating 2D heat equation solver.""" print("=== 2D Heat Equation Solver Example ===\n")

# Create grid - smaller for faster computation
params = he2.create_2d_grid(Lx=1.0, Ly=1.0, T=0.02, Nx=30, Ny=30, Nt=200, D=0.01)

print(f"Grid: {params.Nx}×{params.Ny}, dt={params.dt:.4f}")
print(f"Stability: rx+ry = {params.rx+params.ry:.3f}")
print()

# Test 1: Step function
print("Test 1: Step function initial condition")
u_step = he2.step_function_2d(params.X, params.Y, x_range=(0.4, 0.6), y_range=(0.4, 0.6))

results_step = he2.compare_2d_methods(u_step, params)
he2.plot_method_comparison_2d(params.X, params.Y, u_step, results_step)

# Test 2: Gaussian pulse
print("\nTest 2: Gaussian pulse initial condition")
u_gauss = he2.gaussian_2d(params.X, params.Y, center=(0.3, 0.7), width=(0.05, 0.08))

# Just test one method for the Gaussian
u_final = he2.solve_crank_nicolson_2d(u_gauss, params)
he2.plot_2d_solution(params.X, params.Y, u_gauss, u_final, "Crank-Nicolson")

# Test 3: Multiple hotspots
print("\nTest 3: Multiple hotspots")
u_multi = (he2.gaussian_2d(params.X, params.Y, (0.3, 0.3), (0.05, 0.05), 1.0) +
           he2.gaussian_2d(params.X, params.Y, (0.7, 0.7), (0.08, 0.05), 0.8) +
           he2.gaussian_2d(params.X, params.Y, (0.3, 0.7), (0.06, 0.08), 1.2))

u_final_multi = he2.solve_backward_euler_2d(u_multi, params)
he2.plot_2d_solution(params.X, params.Y, u_multi, u_final_multi, "Backward Euler")

print("\n=== All tests completed! ===")

if name == "main": main() ```

Exercises

Exercise 1. For a 2D step function on \([0.4, 0.6] \times [0.4, 0.6]\) with \(D = 0.01\) and \(T = 0.02\), describe qualitatively how the temperature profile evolves. Does the heated region remain rectangular?

Solution to Exercise 1

At \(t = 0\), the temperature is 1 inside the square \([0.4, 0.6]^2\) and 0 outside. As time progresses, heat diffuses outward in all directions equally. The sharp rectangular boundary smooths into a rounded profile because the corners of the rectangle have the highest curvature, causing the fastest diffusion there.

By \(t = 0.02\), the profile resembles a smoothed bump that is nearly circular (for a square initial condition on a square domain). The rectangular shape is lost because corners diffuse faster than edges, and the characteristic diffusion length \(\sqrt{4DT} = \sqrt{0.0008} \approx 0.028\) is comparable to the initial patch size.

Exercise 2. The code uses compare_2d_methods to compare Forward Euler, Backward Euler, Crank-Nicolson, and ADI. Rank these methods by expected accuracy for the given parameters.

Solution to Exercise 2

For \(N_x = N_y = 30\), \(N_t = 200\), \(D = 0.01\), \(T = 0.02\): \(r_x + r_y \approx 0.024\), well within the stability limit.

Expected accuracy ranking (best to worst):

  1. Crank-Nicolson: \(O(\Delta t^2 + \Delta x^2 + \Delta y^2)\)
  2. ADI: Second-order in time and space (similar to CN but with splitting error)
  3. Forward Euler: \(O(\Delta t + \Delta x^2 + \Delta y^2)\)
  4. Backward Euler: \(O(\Delta t + \Delta x^2 + \Delta y^2)\) (same order as FE but with more numerical diffusion)

In practice, CN and ADI produce nearly identical results, and both significantly outperform the first-order methods for this problem.

Exercise 3. Why does the code create a multi-hotspot initial condition by summing three Gaussians? Is the result an exact solution to any PDE?

Solution to Exercise 3

The code sums three Gaussians to model a physically motivated scenario: multiple localized heat sources at different positions, intensities, and widths. The sum is simply a superposition of the individual Gaussians.

This initial condition is not itself a solution to any simple PDE, but because the heat equation is linear, the solution at time \(t\) is the sum of the individual Gaussian evolutions. Each Gaussian broadens independently as \(\sigma_i(t) = \sqrt{\sigma_{i,0}^2 + 2Dt}\) and reduces in amplitude accordingly.

Exercise 4. The Backward Euler method is used for the multiple hotspots test. Explain why this choice is robust for production use, even though it is only first-order in time.

Solution to Exercise 4

Backward Euler is robust because:

  1. Unconditional stability: No time-step restriction, so the user can choose \(\Delta t\) based on accuracy requirements rather than stability constraints.
  2. Damping of high frequencies: Numerical diffusion smooths out high-frequency oscillations that might arise from rough initial conditions.
  3. Reliable convergence: The sparse linear system always has a unique solution (the matrix is positive definite).
  4. Monotonicity: The solution preserves non-negativity, which is physically correct for temperature.

For production code where robustness matters more than optimal accuracy, Backward Euler is a safe default. For higher accuracy, one would switch to Crank-Nicolson.