Linear Advection-Diffusion Equation

This chapter provides a rigorous exploration of the linear advection–diffusion equation and its numerical treatment, emphasizing its central role in modeling advection-diffusion equation. It begins with the formulation of the 1D advection–diffusion equation and proceeds through spatial and temporal discretization using central differences and explicit Euler methods. Von Neumann stability analysis is applied to derive the amplification factor, highlighting the influence of key dimensionless numbers: Courant number (CFL), and diffusion number (σ). These parameters guide the conditions under which the solution remains stable. The chapter also discusses the role of the cell Reynolds number which is crucial to keep solution free from spurious oscillations. Advanced schemes such as QUICK are discussed for its improved dispersion–dissipation behavior. Learners gain tools to design accurate, stable solvers for convection–diffusion problems.