Upwind scheme explained

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.^[1]

Model equation

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

	\partialu
	\partialt

	\partialu
	\partialx

which describes a wave propagating along the

-axis with a velocity

. This equation is also a mathematical model for one-dimensional linear advection. Consider a typical grid point

in thedomain. In a one-dimensional domain, there are only two directions associated with point

– left (towards negative infinity) andright (towards positive infinity). If

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 side. Similarly, if

is negative the traveling wave solution propagates towards the left, the left side is called downwind side and right side is the upwind side. If the finite difference scheme for the spatial derivative,

\partialu/\partialx

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]

where

refers to the

dimension and

refers to the

dimension. (By comparison, a central difference scheme in this scenario would look like

n+1

	n
u
	i

\Deltat

	n
u
	i-1

i+1

2\Deltax

=0,

regardless of the sign of

Compact form

Defining

a⁺=max(a,0), a^-=min(a,0)

and

	-
u
	x

	n
u
	i-1

\Deltax

	+
u
	x

	n
u
	i

i+1

\Deltax

the two conditional equations and can be combined and written in a compact form asEquation (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.^[3]

c=\left|

	a\Deltat
	\Deltax

\right|\le1

and

0\lea

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

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,

	-
u
	x

becomes the 3-point backward difference in equation and is defined as

	-
u
	x

	n
4u
	i-1

	n
u
	i-2

2\Deltax

and

	+
u
	x

is the 3-point forward difference, defined as

	+
u
	x

-u

	n
4u
	i+1

	n
3u
	i

i+2

2\Deltax

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

Notes and References

Courant. Richard. Isaacson. E. Rees. M.. On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Differences. Comm. Pure Appl. Math.. 1952. 5. 3. 243..255. 10.1002/cpa.3160050303.
Book: Patankar, S. V. . Suhas Patankar
. Suhas Patankar . Numerical Heat Transfer and Fluid Flow . . 1980 . 978-0-89116-522-4.
Book: Hirsch, C.. Numerical Computation of Internal and External Flows. 1990. . 978-0-471-92452-4.

Upwind scheme explained

Model equation

First-order upwind scheme

Compact form

Stability

Second-order upwind scheme

See also

Notes and References