Upwind Differencing and Artificial Viscosity

Consider the model Burger's equation. (9.11) and focus the attention on the inertia terms

As seen, the simple upwind scheme gives

for  u > 0

for  u < 0

From Taylor series expansion, we can write

(11.8)

(11.9)

Substituting Eqns. (11.8) and (11.9) into (11.3) gives (dropping the subscript i and superscript n)

[Diffusive terms]

or

which may be rewritten as

higher order terms

(11.10)

where

C (Courant number)

In deriving Eq. (11.10), was taken as .

However, the nonphysical coefficient leads to diffusion like term which is dependent on the discretization procedure. This is known as the numerical or artificial viscosity.

Let us look at the expression somewhat more critically..

, for u > 0

(11.11)

On one hand we have considered that u > 0 and on the other CFL condition demands that C < 1 (so that the algorithm can work).

As a consequence, is always a positive non-zero quantity ( so that the algorithm can work).

If, instead of analyzing the transient equation, we put in Eq. (11.3) and expand it in Taylor series, we obtain

(11.12)