Numerical Methods for Fokker–Planck Equations I

Electrons in a device, particles in a double well, a population of firing neurons, gene frequencies drifting to fixation, and the noising of data in a diffusion model all move probability by the same law. A Fokker–Planck equation keeps mass, keeps its density positive, holds an equilibrium and lets a relative entropy only decrease. A scheme that loses one of these is wrong, however small its error.

This first volume derives the equation from its stochastic differential equation, sets out five running examples with their exact solutions, and states the structure a scheme must keep. It then builds the schemes that keep it: finite volumes with Scharfetter–Gummel and Chang–Cooper fluxes, time steps that stay positive at any size, and the kinetic equation in phase space with its Gibbs density held exactly. Where the structure-keeping scheme is the less accurate one, the book measures by how much.

Every main result is proved. Every figure and every number comes from the Python printed whole in the appendix, checked against an exact solution and computed two ways.

Sanne Kooistra is a numerical analyst from Delft who has simulated electrons in semiconductors and fusion plasmas and the firing of neuron populations, and has counted the mass at every step.