2D Unsteady Parabolic Problems

This chapter discusses 2D unsteady parabolic problems, with a focus on solving the heat equation using both explicit and implicit numerical schemes. It begins with a formulation of the heat equation and introduces the Gaussian pulse as a localized initial condition, followed by time-dependent Dirichlet boundary conditions. Spatial derivatives are approximated using second-order central differences (FDO2), and explicit time integration is handled via the Euler method under strict stability constraints. To overcome the limitations of small time steps in explicit schemes, the Alternating Direction Implicit (ADI) method is introduced. ADI decouples the 2D problem into one-dimensional implicit steps, offering enhanced stability and efficiency. The chapter also presents the Successive Over-Relaxation (SOR) method as an iterative solver for implicit systems, emphasizing its convergence behavior based on the relaxation parameter.