Upwind scheme on arvau

Wikipedia / Upwind scheme

In computational physics, the term advection scheme refers to a class of numerical discretization methods for solving hyperbolic partial differential equations. In the so-called upwind schemes typically, the so-called upstream variables are used to calculate the derivatives in a flow field. That is, derivatives are estimated using a set of data points biased to be more "upwind" of the query point, with respect to the direction of the flow. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Rees who proposed the CIR method.

Model equation

To illustrate the method, consider the following one-dimensional linear advection equation. This equation is also a mathematical model for one-dimensional linear advection is positive, the traveling wave solution of the equation above propagates towards the right, the left side is called the upwind side and the right side is the downwind is negative the traveling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind contains more points in the upwind side, the scheme is called an upwind-biased or simply an upwind scheme.

First-order upwind scheme

The simplest upwind scheme possible is the first-order upwind scheme. It is given by

Compact form

the two conditional equations () and () can be combined and written in a compact form as Equation (3) is a general way of writing any upwind-type schemes.

Stability

The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL) is satisfied.

A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and time. Modified wavenumber analysis shows that the first-order upwind scheme introduces severe numerical diffusion/dissipation in the solution where large gradients exist due to necessity of high wavenumbers to represent sharp gradients.

Second-order upwind scheme

becomes the 3-point backward difference in equation ( This scheme is less diffusive compared to the first-order accurate scheme and is called linear upwind differencing (LUD) scheme.