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Upwind scheme 1

Upwind scheme

In computational fluid dynamics, upwind schemes denote a class of numerical discretization methods for solving

hyperbolic partial differential equations. Upwind schemes use an adaptive or solution-sensitive finite difference

stencil to numerically simulate more properly the direction of propagation of information in a flow field. The 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 Rees who proposed the CIR method.[1]

Model equation

To illustrate the method, consider the following one-dimensional linear wave equation

It describes a wave propagating in the -direction with a velocity . The preceding equation is also a

mathematical model for one-dimensional linear advection. Consider a typical grid point in the domain. In aone-dimensional domain, there are only two directions associated with point - left and right. If is positive the

left side is called upwind side and right side is the downwind side. Similarly, if is negative the left side is called

downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivative,

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[2]

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 upwind-type schemes. The upwind scheme is stable if the following

CourantFriedrichsLewy condition (CFL) condition is satisfied.[3]

A Taylor series analysis of the upwind scheme discussed above will show that it is first-order accurate in space and

time. The first-order upwind scheme introduces severe numerical diffusion in the solution where large gradients

exists.
http://en.wikipedia.org/w/index.php?title=Numerical_diffusionhttp://en.wikipedia.org/w/index.php?title=Taylor_serieshttp://en.wikipedia.org/w/index.php?title=Courant%E2%80%93Friedrichs%E2%80%93Lewy_conditionhttp://en.wikipedia.org/w/index.php?title=Numerical_stabilityhttp://en.wikipedia.org/w/index.php?title=Richard_Couranthttp://en.wikipedia.org/w/index.php?title=Finite_differencehttp://en.wikipedia.org/w/index.php?title=Hyperbolic_partial_differential_equationhttp://en.wikipedia.org/w/index.php?title=Discretizationhttp://en.wikipedia.org/w/index.php?title=Computational_fluid_dynamics


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Upwind scheme 2

Second-order upwind scheme

The spatial accuracy of the first-order upwind scheme can be improved by choosing a more accurate finite difference

stencil for the approximation of spatial derivative. For the second-order upwind scheme, in equation (3) is

defined as

and is defined as

This scheme is less diffusive compared to the first-order accurate scheme and is called linear upwind differencing

(LUD) scheme.

Third-order upwind scheme

For the third-order upwind scheme, in equation (3) is defined as

and is defined as

This scheme is less diffusive compared to the second-order accurate scheme. However, it is known to introduce

slight dispersive errors in the region where the gradient is high.

References

[1] Courant, R., Isaacson, E., and Rees, M. (1952). "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences",Comm. Pure Appl. Math., 5, 243-255.

[2] Patankar, S. V. (1980).Numerical Heat Transfer and Fluid Flow. Taylor & Francis. ISBN 978-0891165224.

[3] Hirsch, C. (1990).Numerical Computation of Internal and External Flows. John Wiley & Sons. ISBN 978-0471924524.
http://en.wikipedia.org/w/index.php?title=John_Wiley_%26_Sonshttp://en.wikipedia.org/w/index.php?title=Taylor_%26_Francishttp://en.wikipedia.org/w/index.php?title=Suhas_Patankarhttp://en.wikipedia.org/w/index.php?title=Richard_Courant


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Article Sources and Contributors 3

Article Sources and ContributorsUpwind scheme Source: http://en.wikipedia.org/w/index.php?oldid=397298341 Contributors: Berland, Eymann, J.Voss, Malosse, Michael Hardy, Moorekwesi, Salih, Stone Violin, Tide rolls,

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