ADI Method

In this this project, we implement a relatively advanced finite difference PDE solver to solve a 2D parabolic equation. Finite difference methods approximate time and spatial derivatives by using finite difference quotients on a discrete grid—transforming the continuous PDE into a system of algebraic equations. One particularly effective scheme for multi‐dimensional parabolic problems is the Alternating Direction Implicit (ADI) method.

The essence of the ADI method is to split the fully implicit time step into a sequence of one‐directional implicit updates, so that at each substep we only have to solve a tridiagonal linear system along one spatial coordinate while treating the other coordinates explicitly.

The key advantage of the ADI method is that it combines unconditional stability (typical of fully implicit schemes) with a computational cost comparable to explicit or semi‐implicit methods. By alternating the implicit direction at each half‐step, we avoid having to invert a large multidimensional matrix, replacing it instead with two one‐dimensional solves. This dimension‐by‐dimension splitting drastically reduces storage requirements and arithmetic complexity. Consequently, ADI schemes allow for larger time‐step sizes without sacrificing stability, making them highly efficient for long‐time integration of parabolic equations on regular grids.

This notebook will serve as an introduction to solving the two‐dimensional heat (or diffusion) equation using the Peaceman–Rachford ADI method. We will discuss the theoretical foundation of the ADI splitting, derive the finite‐difference stencils, and outline the implementation details in MATLAB.