In this chapter we consider numerical methods for solving parabolic equations. The canonical equation of this type is given by

∂u

∂t = ∇ ·D · ∇u. (7.1)

This is sometimes called the heat equation as it describes the conduction of heat throughout a material. It is also prototypical of many types of diffusive processes that occur in a magnetized plasma, including the transport of heat, particles, angular momentum, and magnetic flux in real space, and the evolution of the distribution function in velocity space due to particle collisions. By allowing the diffusion coefficient D in Eq. (7.1) to be a tensor, we can describe anisotropic processes such as the diffusion of heat and particles when a magnetic field is present. In the next section we discuss the basic numerical algorithms for solving equations of the parabolic type in one dimension, such as the surface-averaged equations derived in the last chapter. We then discuss extension of these methods to multiple dimensions.