2D Advection-Diffusion Solver
Serial and parallel solvers for the 2D advection-diffusion equation using Python
This project implements serial and parallel solvers for the 2D Advection-Diffusion Equation, a fundamental PDE describing how a scalar quantity (e.g., temperature, concentration) evolves under the combined influence of advection (transport) and diffusion (spreading).
Physics Background
In 2D, the general form of the advection-diffusion equation is:
\[\frac{\partial \phi}{\partial t} + u \frac{\partial \phi}{\partial x} + v \frac{\partial \phi}{\partial y} = D \left( \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} \right)\]Where:
- ( phi(x, y, t) ): scalar field (e.g., temperature)
- ( u, v ): advection velocities
- ( D ): diffusion coefficient
- ( x, y ): spatial coordinates
- ( t ): time
Problem Setup
For this implementation:
- Advection is neglected (but it can be varied in the code) → ( u = v = 0 )
- The equation simplifies to pure diffusion:
- Boundary conditions: Dirichlet (fixed values) on all sides
- Initial condition: Typically a Gaussian pulse or localized concentration in the center
- Time integration: Explicit Euler scheme
Scalability and Performance Results
The following table summarizes the execution times and speedups for various domain decompositions:
| Configuration | Simulation Time (seconds) | Speedup (relative to serial) |
|---|---|---|
| Serial | 1346.69 | 1.00× |
| Parallel (MPI Decomposition: 1×2) | 773.87 | 1.74× |
| Parallel (MPI Decomposition: 2×2) | 442.39 | 3.04× |
| Parallel (MPI Decomposition: 1×5) | 403.66 | 3.33× |
Results Discussion
Solution Consistency
- Both the serial and parallel solvers produce nearly identical solution fields, verifying correctness.
- Contour plots from each implementation confirm spatial consistency.
Scalability
- Clear performance gains are observed in the parallel version:
- 2 processors → 1.74× faster
- 4 processors → 3.04× faster
- 5 processors → 3.33× faster
MPI Decomposition Impact
- Decomposing the domain as 2×2 yields better load balancing and communication efficiency.
- Unbalanced splits like 1×5 still perform well but may incur higher communication overhead.
Performance Trade-Offs
- For small grids, MPI communication overhead can dominate.
- For larger domains, parallel scaling becomes more effective and efficient.
Solution Plot:
