# Exponential and Power Law Schemes

Since the exact solution of eq. (4.201) exists, one can reasonably expect that an accurate scheme can be derived if the result of the exact solution, eq. (4.210), is utilized. Equation (4.201) can be rewritten as

 $\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0$ (1)

Defining the total flux of $\varphi$ due to convection and diffusion

 $J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}$ (2)

eq. (4.229) becomes

 $\frac{dJ}{dx}=0$ (3)

Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields

 $\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}$ (4)

Instead of assuming piecewise linear distribution of $\varphi$ as with central difference scheme or assuming $\varphi$ at the face of the control volume is equal to the value of $\varphi$ at the grid point on the upwind side in the upwind scheme, the distribution of $\varphi$ between grid points can be taken as that obtained from the exact solution, eq. (4.210). Applying eq. (4.210) between grid points E and P, we have

 $\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}$ (5)

Substituting eq. (4.233) into eq. (4.230) and evaluating the result at x = xe, the total flux of $\varphi$ at the face of control volume becomes

 $J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]$ (6)

Similarly, the total flux at the west face of the control volume is

 $J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]$ (7)

Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields

 $a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}$ (8)

where

 $a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}$ (9)
 $a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}$ (10)
 $\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})$ (11)

Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,

 $a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}$ (12)
 $a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}$ (13)

The comparison of aE / De for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.

Figure 1: Comparison among different schemes.

While the exponential scheme is accurate, the computational time is much longer than for the central difference, upwind or hybrid schemes. Patankar (1981) proposed a power law scheme that has almost the same accuracy as the exponential scheme but a substantially shorter computational time. The coefficient of the neighbor grid point on the east side can be obtained by

$a_{E}/D_{e}=\left\{ \begin{matrix} -\text{Pe}_{\Delta e}\text{ Pe}_{\Delta e}<-10 \\ (1+0.1\text{Pe}_{\Delta e})^{5}-\text{Pe}_{\Delta e}\text{ }-1\text{0}\le \text{Pe}_{\Delta e}<0 \\ (1-0.1\text{Pe}_{\Delta e})^{5}\text{ 0}\le \text{Pe}_{\Delta e}\le 10 \\ 0\text{ Pe}_{\Delta e}>10 \\ \end{matrix} \right.$

which can be rewritten in the following compact form

 $a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]$ (14)