Parallelized Diffusion Simulation Using MPI

This project implements a scalable 2D diffusion simulation using the Message Passing Interface (MPI) to accelerate scientific computation. The simulation models how substances diffuse over time using finite difference approximations and is designed to efficiently run on high-performance computing clusters.

---

Features

• Implicit Euler method for numerical stability with larger time steps
• Conjugate Gradient Solver for sparse linear systems
• Neumann (zero-flux) boundary conditions
• Ghost cell communication between subgrids using MPI_Sendrecv
• Strong and Weak Scaling studies up to 64 processors
• Profiling analysis for performance bottlenecks

---

Background

Diffusion is a fundamental physical process that occurs in many scientific fields, such as physics, biology, and chemistry. However, simulating diffusion over large, high-resolution grids is computationally expensive. A serial implementation becomes impractical at scale.

This project parallelizes the simulation using MPI, distributing the workload across multiple processors and significantly reducing runtime.

---

Implementation Overview

• Discretized the 2D diffusion equation using finite difference methods.
• Used implicit Euler to ensure numerical stability.
• Solved the resulting linear system at each time step using the Conjugate Gradient method.
• Parallelized matrix-vector multiplication and grid updates with MPI.
• Used ghost cells and MPI_Allreduce for process synchronization and boundary updates.

---

Performance

Strong Scaling

• Fixed grid size: 1000 x 1000
• Processors: 1, 2, 4, 8, 16, 32
• Achieved near-linear speedup up to 8 processors.
• Speedup plateaus at 16+ due to communication overhead.

Weak Scaling

• Scaled problem size with processor count.
• Runtime grows as processor count increases, due to ghost cell communication and load imbalance.

---

Hardware & Parameters

• Tested on Perlmutter CPU nodes (128 cores per node)
• Diffusion coefficient: D = 0.1
• Time step size: dt = 0.001
• Iterations: 200
• Domain size: 1.0 x 1.0

---

Profiling

• Major runtime bottleneck: matrix-vector product in CG solver.
• MPI profiling shows imbalanced workload across ranks, especially in MPI_Allreduce and MPI_Sendrecv.

Authors

• Thomas Cui (zc246)
• Bohan Yang (by93)
• Jingyu Xu (jx62)

---