Notes on Diffusion

These summarize methods for solving the diffusion equation. 1 Parabolic equations The diffusion equation is ∂φ ∂t = ∂ ∂x k ∂φ ∂x (1) This can describe thermal diffusion (for example, as part of the energy equation in compressible flow), species/mass diffusion for multispecies flows, or the viscous terms in incompressible flows. In this form, the diffusion coefficient (or conductivity), k, can be a function of x, or even φ. We will consider a constant diffusion coefficient: ∂φ ∂t = k ∂ 2 φ ∂x 2 (2) The diffusion equation is the prototypical parabolic PDE. The basic behavior of the diffusion equation is to take strongly peaked concentrations of φ and smooth them out with time. The simplest way to difference this equation is explicit in time (i.e. the righthand side depends only on the old state): φ n+1 i − φ n i ∆t = k φ n i+1 − 2φ n i + φ n i−1 ∆x 2 (3) This is second-order accurate in space, but only first order accurate in time (since the righthand side is not centered in time). As with the advection equation, when differenced explicitly, there is a constraint on the timestep required for stability. Looking at the growth of a single Fourier mode, φ = A n e ijθ with j = √ −1, we find: A n+1