Solving Partial Differential Equations Using the ADI Method (Two-Micro Heater Equation)

The ADI (Alternating Direction Implicit) method proves particularly effective for solving partial differential equations like the two-dimensional heating equation. This numerical approach decomposes multidimensional PDEs into sequential one-dimensional sub-problems through operator splitting. By alternating between implicit solutions along x and y directions, the method maintains stability while reducing computational complexity from O(N²) to O(N) per dimension. Key implementation steps include: 1. Discretizing the spatial domain using finite differences and applying the Crank-Nicolson scheme for temporal integration 2. Solving tridiagonal systems efficiently with Thomas algorithm in each alternating direction 3. Implementing dimensional splitting with appropriate boundary condition handling The ADI method's matrix formulation involves solving (I - ½ΔtA_x)T* = (I + ½ΔtA_y)T^n for the first half-step, followed by (I - ½ΔtA_y)T^{n+1} = (I + ½ΔtA_x)T* for the second half-step, where A_x and A_y represent discrete differential operators. This technique demonstrates particular advantages in heat transfer simulations, fluid dynamics, and quantum mechanical problems where traditional explicit methods face severe stability constraints. The method's unconditional stability for linear problems makes it suitable for large-scale simulations requiring prolonged time integration.