en>Guy vandegrift at 19:40, 31 December 2013

2013-12-31T19:40:44Z

New page

In [[computational physics]], '''upwind schemes''' denote a class of numerical [[discretization]] methods for solving [[hyperbolic partial differential equation]]s. Upwind schemes use an adaptive or solution-sensitive [[finite difference]] stencil to numerically simulate 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 [[Richard Courant|Courant]], Isaacson, and Rees who proposed the CIR method.<ref>[[Richard Courant|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.</ref>

==Model equation==
To illustrate the method, consider the following one-dimensional linear [[wave equation]]
:<math>
\qquad \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0
</math>
which describes a wave propagating in the <math>x</math>-direction with a velocity <math>a</math>. This equation
is also a mathematical model for one-dimensional linear [[advection]]. Consider a typical grid point <math>i</math> in the
domain. In a one-dimensional domain, there are only two directions associated with point <math>i</math> – left and
right. If <math>a</math> is positive the left side is called ''upwind'' side and right side is the ''downwind'' side. Similarly, if <math>a</math> 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, <math>\partial u / \partial x</math> 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<ref>{{cite book |last=Patankar |first=S. V. |authorlink=Suhas Patankar |title=Numerical Heat Transfer and Fluid Flow |publisher=[[Taylor & Francis]] |year=1980 |isbn=978-0-89116-522-4}}
</ref>
:<math>
\quad (1) \qquad \frac{u_i^{n+1} - u_i^n}{\Delta t} + a \frac{u_i^n - u_{i-1}^n}{\Delta x} = 0 \quad \text{for} \quad a > 0
</math>
:<math>
\quad (2) \qquad \frac{u_i^{n+1} - u_i^n}{\Delta t} + a \frac{u_{i+1}^n - u_i^n}{\Delta x} = 0 \quad \text{for} \quad a < 0
</math>

=== Compact form ===
Defining
:<math>
\qquad \qquad a^+ = \text{max}(a,0)\,, \qquad a^- = \text{min}(a,0)
</math>
and
:<math>
\qquad \qquad u_x^- = \frac{u_i^{n} - u_{i-1}^{n}}{\Delta x}\,, \qquad u_x^+ = \frac{u_{i+1}^{n} - u_{i}^{n}}{\Delta x}
</math>
the two conditional equations (1) and (2) can be combined and written in a compact form as
:<math>
\quad (3) \qquad u_i^{n+1} = u_i^n - \Delta t \left[ a^+ u_x^- + a^- u_x^+ \right]
</math>
Equation (3) is a general way of writing any upwind-type schemes.

=== Stability ===
The upwind scheme is [[numerical stability|stable]] if the following [[Courant–Friedrichs–Lewy condition]] (CFL) condition is satisfied.<ref>{{cite book |last=Hirsch |first=C.|title=Numerical Computation of Internal and External Flows|year=1990|publisher=[[John Wiley & Sons]] |isbn=978-0-471-92452-4}}
</ref>
:<math>
\qquad \qquad c = \left| \frac{a\Delta t}{\Delta x} \right| \le 1 .
</math>

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 exist{{Dubious|date=January 2012}}.

==Second-order upwind scheme==
The spatial accuracy of the first-order upwind scheme can be improved by including 3 data points instead of just 2, which offers a more accurate finite difference stencil for the approximation of spatial derivative. For the second-order upwind scheme, <math>u_x^-</math> becomes the 3-point backward difference in equation (3) and is defined as
:<math>
\qquad \qquad u_x^- = \frac{3u_i^n - 4u_{i-1}^n + u_{i-2}^n}{2\Delta x}
</math>
and <math>u_x^+</math> is the 3-point forward difference, defined as
:<math>
\qquad \qquad u_x^+ = \frac{-u_{i+2}^n + 4u_{i+1}^n - 3u_i^n}{2\Delta x}
</math>
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, <math>u_x^-</math> in equation (3) is defined as
:<math>
\qquad \qquad u_x^- = \frac{2u_{i+1} + 3u_i - 6u_{i-1} + u_{i-2}}{6\Delta x}
</math>
and <math>u_x^+</math> is defined as
:<math>
\qquad \qquad u_x^+ = \frac{-u_{i+2} + 6u_{i+1} - 3u_i - 2u_{i-1}}{6\Delta x}
</math>
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.

==See also==
* [[Finite difference method]]
* [[Upwind differencing scheme for convection]]

==References==
{{reflist}}

{{Numerical PDE}}

{{DEFAULTSORT:Upwind Scheme}}
[[Category:Computational fluid dynamics]]
[[Category:Numerical differential equations]]

Free expansion - Revision history

en>Guy vandegrift at 19:40, 31 December 2013