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Upwind schemeIn computational fluid dynamics, the upwind schemes are any of a class of discretization methods to solve hyperbolic partial differential equations numerically. The wave eation, the advection equation, the Euler equations in fluid dynamics, etc. belongs to hyperbolic PDEs. Upwind schemes use an adaptive or solutionsensitive finite difference stencil to numerically simulate more properly the direction of propagation of information in a flow field. More specifically, upwind schemes attempt to discretize hyperbolic partial differential equations by using differencing biased in the direction determined by the sign of the characteristic speeds. Historically, the origin of upwind methods can be traced back to the work of Courant, Isaacson, and Reeves who proposed the CIR method. Additional recommended knowledge
Model equationTo illustrate the method, consider the following onedimensional linear wave equation It describes a wave propagating in the xdirection with a velocity a. The preceding equation is also a mathematical model for onedimensional linear advection. Consider a typical grid point i in the domain. In a onedimensional domain, there are only two direction associated with point i  left and right. If a is positive the left side is called upwind side and right side is the downwind side. Similarly, if a is negative the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the derivative, contain more points in the upwind side, the scheme is called an upwindbiased or simply upwind scheme. Firstorder upwind schemeThe simplest upwind scheme possible is the firstorder upwind scheme. It is given by Defining and the two conditional equations (1) and (2) can be combined and written in a compact form as Equation (3) is a general way of writing any upwindtype schemes. The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL) condition is satisfied. A Taylor series analysis of the upwind scheme discussed above will show that it is firstorder accurate in space and time. The firstorder upwind scheme introduces severe numerical diffusion in the solution where large gradients exists. Secondorder upwind schemeThe spatial accuracy of the firstorder upwind scheme can be improved by choosing a more accurate finite difference stencil for the approximation of spatial derivative. For the secondorder upwind scheme, in equation (3) is defined as and is defined as This scheme is less diffusive compared to the firstorder accuarte scheme. Thirdorder upwind schemeFor the thirdorder upwind scheme, in equation (3) is defined as and is defined as This scheme is less diffusive compared to the secondorder accuarte scheme. However, it is known to introduce slight dispersive errors in the region where the gradient is high. References


This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Upwind_scheme". A list of authors is available in Wikipedia. 