Numerical Solutions for Partial Differential Equations - Implementation and Comparative Analysis

This laboratory report presents a comprehensive study on numerical solutions for partial differential equations. We begin by implementing the classical explicit difference scheme to solve initial-boundary value problems for two-dimensional diffusion equations, featuring a straightforward finite difference discretization approach where the solution at each grid point is computed using adjacent values from the previous time step. Subsequently, we apply the Peaceman-Rachford (P-R) alternating direction implicit (ADI) difference scheme, which employs a splitting technique to handle spatial dimensions separately through tridiagonal matrix algorithms, providing improved stability compared to explicit methods. Finally, we implement the locally one-dimensional (LOD) format, another operator splitting method that decomposes the multidimensional problem into sequential one-dimensional solves. The implementation details include matrix assembly procedures, time-step selection criteria, and boundary condition handling mechanisms. To demonstrate our experimental findings comprehensively, we provide complete source code with detailed comments, runtime results showing convergence behavior and computational efficiency, and a systematic comparison analysis of these algorithms focusing on stability conditions, accuracy metrics, and computational complexity to facilitate deeper understanding of numerical methods for partial differential equations.