Diffusion Problem

This chapter presents a comprehensive study of the diffusion equation, highlighting both physical and numerical aspects. It begins by distinguishing physical diffusion from numerical diffusion, emphasizing the importance of accurately representing the diffusion coefficient in simulations. Through Fourier analysis, the chapter introduces the continuum response function and key non-dimensional parameters like k ∗ and the diffusion number σ. Several numerical schemes are developed and analyzed explicit FTCS, leapfrog, DuFort–Frankel, Crank–Nicolson, and Laasonen for example. Each of these schemes are evaluated for stability, accuracy, and damping behavior. Stability criteria are derived via amplification factors, revealing that explicit methods require strict timestep limits, while implicit schemes offer unconditional stability. Modal analysis via matrix decomposition highlights eigenvalue-driven decay rates and stiffness in discretized diffusion problems.