In computational physics, upwind schemes denote a class of numerical discretization methods for solving hyperbolic partial differential equations. Upwind schemes use an adaptive or solutionsensitive 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 Courant, Isaacson, and Rees who proposed the CIR method.^{[1]}
Contents

Model equation 1

Firstorder upwind scheme 2

Compact form 2.1

Stability 2.2

Secondorder upwind scheme 3

Thirdorder upwind scheme 4

See also 5

References 6
Model equation
To illustrate the method, consider the following onedimensional linear advection equation

\qquad \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0
which describes a wave propagating along the xaxis with a velocity a. This 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 directions 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 spatial derivative, \partial u / \partial x contains more points in the upwind side, the scheme is called an upwindbiased or simply an upwind scheme.
Firstorder upwind scheme
The simplest upwind scheme possible is the firstorder upwind scheme. It is given by^{[2]}

\quad (1) \qquad \frac{u_i^{n+1}  u_i^n}{\Delta t} + a \frac{u_i^n  u_{i1}^n}{\Delta x} = 0 \quad \text{for} \quad a > 0

\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
Compact form
Defining

\qquad \qquad a^+ = \text{max}(a,0)\,, \qquad a^ = \text{min}(a,0)
and

\qquad \qquad u_x^ = \frac{u_i^{n}  u_{i1}^{n}}{\Delta x}\,, \qquad u_x^+ = \frac{u_{i+1}^{n}  u_{i}^{n}}{\Delta x}
the two conditional equations (1) and (2) can be combined and written in a compact form as

\quad (3) \qquad u_i^{n+1} = u_i^n  \Delta t \left[ a^+ u_x^ + a^ u_x^+ \right]
Equation (3) is a general way of writing any upwindtype schemes.
Stability
The upwind scheme is stable if the following Courant–Friedrichs–Lewy condition (CFL) condition is satisfied.^{[3]}

\qquad \qquad c = \left \frac{a\Delta t}{\Delta x} \right \le 1 .
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 exist.
Secondorder upwind scheme
The spatial accuracy of the firstorder 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 secondorder upwind scheme, u_x^ becomes the 3point backward difference in equation (3) and is defined as

\qquad \qquad u_x^ = \frac{3u_i^n  4u_{i1}^n + u_{i2}^n}{2\Delta x}
and u_x^+ is the 3point forward difference, defined as

\qquad \qquad u_x^+ = \frac{u_{i+2}^n + 4u_{i+1}^n  3u_i^n}{2\Delta x}
This scheme is less diffusive compared to the firstorder accurate scheme and is called linear upwind differencing (LUD) scheme.
Thirdorder upwind scheme
For the thirdorder upwind scheme, u_x^ in equation (3) is defined as

\qquad \qquad u_x^ = \frac{2u_{i+1} + 3u_i  6u_{i1} + u_{i2}}{6\Delta x}
and u_x^+ is defined as

\qquad \qquad u_x^+ = \frac{u_{i+2} + 6u_{i+1}  3u_i  2u_{i1}}{6\Delta x}
This scheme is less diffusive compared to the secondorder accurate scheme. However, it is known to introduce slight dispersive errors in the region where the gradient is high.
See also
References

^ Courant, Richard; Isaacson, E; Rees, M. (1952). "On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences". Comm. Pure Appl. Math. 5: 243..255.

^

^ Hirsch, C. (1990). Numerical Computation of Internal and External Flows.
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