2D Heat Diffusion Equation Solver

Overview

This project implements a numerical solver for the 2D stationary heat diffusion equation on a rectangular domain. It uses the Finite Difference Method (FDM) to discretize the Partial Differential Equation (PDE) into a sparse linear system, which is then solved efficiently.

A key feature is the custom implementation of 2D quadratic spline interpolation, built from scratch using successive 1D row and column interpolations. The coefficients for these splines are calculated using a custom QR factorization algorithm (via modified Gram-Schmidt), also implemented from scratch.

The solver is validated against an analytical solution to analyze convergence rates across different grid densities.

Key Features

• Finite Difference Method (FDM): Discretizes the 2D PDE with variable diffusion coefficients into a large, sparse linear system.
• Custom Spline Interpolation: Implements 2D quadratic spline interpolation from scratch to reconstruct a continuous solution from discrete grid points.
• QR Factorization from Scratch: Solves the linear systems for spline coefficients using a custom-built QR decomposition algorithm.
• Sparse Matrix Operations: Utilizes scipy.sparse for efficient storage and solution of the large linear systems arising from FDM.
• Error & Convergence Analysis: Automatically compares numerical results with an analytical solution, plotting error surfaces and calculating convergence rates.

Mathematical Model

The solver addresses the stationary heat diffusion equation: $$- \nabla \cdot (k(x,y) \nabla u) = f(x,y)$$ on the domain $\Omega = [0, 5] \times [0, 2]$, subject to:

• Dirichlet Boundary Conditions: On $y=0$ and $y=2$.
• Neumann Boundary Conditions: On $x=0$ and $x=5$.

Libraries Used

• NumPy: For numerical operations and dense matrix handling in the QR solver.
• SciPy: Specifically scipy.sparse and scipy.sparse.linalg.spsolve for handling the main FDM linear system.
• Matplotlib: For 3D visualization of solutions and error surfaces, and 2D convergence plots.

How to Run

1. Ensure you are in the root directory of the project:

   pip install -r requirements.txt

2. Run the main script:

   python main.py

The script will sequentially solve the problem for grid sizes $N \in {4, 8, 16, 32}$, display 3D plots for each, and finally show a convergence graph with calculated rates.