A Primer on Stochastic Partial Differential Equations with Spatially Correlated Noise

Abstract

With the growing number of microscale devices from computer memory to microelectromechanical systems, such as lab-on-a-chip biosensors and the increased ability to experimentally measure at the micro- and nanoscale, modeling systems with stochastic processes is a growing need across science. In particular, stochastic partial differential equations (SPDEs) naturally arise from continuum models—for example, a pillar magnet's magnetization or an elastic membrane's mechanical deflection. In this review, I seek to acquaint the reader with SPDEs from the point of view of numerically simulating their finite-difference approximations, without the rigorous mathematical details of assigning probability measures to the random field solutions. I will stress that these simulations with spatially uncorrelated noise may not converge as the grid size goes to zero in the way that one expects from deterministic convergence of numerical schemes in two or more spatial dimensions. I then present some models with spatially correlated noise that maintain sampling of the physically relevant equilibrium distribution. Numerical simulations are presented to demonstrate the dynamics; the code is publicly available on GitHub.