Solving Laplace's Equation using the Five-Point Finite Difference Method

This program utilizes the five-point finite difference scheme to solve Laplace's equation. Laplace's equation finds widespread applications in various scientific and engineering domains. The five-point finite difference method is a high-accuracy numerical technique for discretizing partial differential equations, commonly employed for obtaining numerical solutions to PDEs. The implementation in MATLAB employs a structured grid approach where the Laplacian operator is approximated using central differences in both x and y directions. Key algorithmic components include constructing the coefficient matrix representing the discrete Laplacian and solving the resulting linear system using efficient numerical methods like Gauss-Seidel iteration or direct matrix solvers. Developed using MATLAB as the primary environment, this program facilitates straightforward numerical simulations and visualization analysis through built-in matrix operations and graphical tools. The code architecture allows parameter adjustments for solving different problem configurations, demonstrating significant flexibility and extensibility. Boundary conditions can be modified through input parameters, and the solution convergence can be monitored through residual calculations. The program structure includes modular functions for grid generation, matrix assembly, solver implementation, and result visualization, making it suitable for educational purposes and practical engineering applications.