Upwind scheme

Upwind schemes are a class of numerical discretisation methods used in computational physics to solve hyperbolic partial differential equations. They estimate derivatives in a flow field using data points biased toward the upwind side of the query point, the side from which the flow is coming, instead of using symmetric neighbours. Historically the origin of upwind methods is traced to the CIR method proposed by Courant, Isaacson and Rees. The simplest is the first-order upwind scheme, illustrated on the one-dimensional linear advection equation. It is stable only when the Courant-Friedrichs-Lewy condition is satisfied and is first-order accurate in space and time, and it introduces severe numerical diffusion where gradients are large. Using three data points instead of two gives the second-order upwind scheme, which offers a more accurate finite difference stencil for the spatial derivative.