This tutorial introduces diffusion processes using both manual finite difference methods and Basilisk's built-in solvers, demonstrated through CO2 dispersion modeling.

In this tutorial, you will learn:

1. Finite Difference Method: Manually implement the 2D diffusion equation using central differences
2. Basilisk Diffusion Solver: Use Basilisk's implicit solver for stable and efficient diffusion
3. Source Terms: Model injection sources in diffusion problems
4. Dynamic Visualization: Create animations of time-evolving concentration fields

Objective: Implement CO2 diffusion using explicit finite difference method.

Key Concepts:

• Explicit time integration for diffusion equation: ∂C/∂t = D∇²C
• Central difference approximation for second derivatives
• Stability condition: Δt ≤ Δx²/(4D)
• Source term implementation (CO2 injection)

Physical Setup:

• Domain: 5m × 5m
• Initial condition: 400 ppm CO2 (atmospheric background)
• Injection source: Pipe at center (radius 0.1m) maintaining 1500 ppm
• Diffusivity coefficient: D = 1.60e-5 m²/s

Mathematical Implementation:

// Second derivatives using central differences
d²C/dx² ≈ (C[i+1,j] - 2C[i,j] + C[i-1,j]) / Δx²
d²C/dy² ≈ (C[i,j+1] - 2C[i,j] + C[i,j-1]) / Δy²

// Time update
C_new = C_old + Δt × D × (d²C/dx² + d²C/dy²)

Run the example:

make Ex1_FDCO2.tst

Question: Is the simulation time long enough to let the CO2 diffuse from the injection point to the edges of the room? is that realistic?

Output: CO2_evo.mp4 showing CO2 spreading from injection point

Objective: Solve the same diffusion problem using Basilisk's implicit solver.

Key Concepts:

• Implicit solver: More stable, allows larger timesteps
• diffusion.h solver usage
• Face vector for diffusion coefficients

Key Advantages of Implicit Solver:

• mostly stable: No strict timestep limitation
• Efficient: Built-in sparse matrix solvers

Solver Usage:

const face vector kappa[] = {D, D};  // Diffusion coefficient in x,y
diffusion(C, dt, kappa);             // Solve: dC/dt = D∇²C

Run the example:

make Ex2_basi.tst

Compare:

• Same physical setup as Exercise 1
• Different numerical method
• Observe stability and accuracy differences
• an example of using external solvers

Output: CO2_field.mp4 showing CO2 concentration evolution

Objective: Extend the model to include three CO2 injection points.

Key Concepts:

• Multiple source terms
• Source interaction and superposition
• Spatial distribution of sources
• Data output for quantitative analysis

Source Configuration:

• Nozzle 1: Center (0, 0)
• Nozzle 2: Left (-1.25, 0)
• Nozzle 3: Right (1.25, 0)
• All maintain 1500 ppm within 0.1m radius

New Feature - Data Export:

event printdata (t = 0; t <= 10.; t += 0.3) {
  // Output concentration along centerline for analysis
  for (double x = -L0/2; x < L0/2; x += L0/N)
    fprintf(fp, "%g %g %g\n", t, x, interpolate(C, x, 0));
}

Run the example:

make Ex3_inject3.tst

Output:

• CO2_field.mp4: Visualization of three plumes merging
• y0.dat: Centerline concentration data for plotting

1. How does the spacing between nozzles affect:

   • Time to reach target concentration?
   • Uniformity of distribution?
   • Peak concentration at the center?

2. Can you achieve >80% coverage (800+ ppm) within 3 seconds? What configuration works best?

3. If you could only use 2 nozzles instead of 3, where would you place them?

1. Linear arrangement: All three along x-axis at different spacings
2. Triangular pattern: Equilateral triangle formation
3. L-shape: Corner coverage strategy
4. Your creative design: What works best?