Hybrid Scheme

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The upwind scheme uses the value of \varphi from the grid point at the upwind side as the value of \varphi at the face of the control volume regardless of the grid Peclet number. While this treatment can yield accurate results for cases with high Peclet number, the result will not be accurate for cases where the grid Peclet number is near zero; for which cases the central difference scheme can produce better results. Spalding (1972) proposed a hybrid scheme that uses the central difference scheme when \left| \text{Pe}_{\Delta } \right|\le 2

and the upwind scheme when 

\left| \text{Pe}_{\Delta } \right|>2 . To observe the difference between the central difference and upwind schemes, the coefficient for the east neighboring grid point, eqs. (4.215) and (4.222), can be rewritten as

a_{E}/D_{e}=1-\frac{1}{2}\text{Pe}_{\Delta e},\text{  Central difference scheme}


a_{E}/D_{e}=1+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right],\text{  Upwind scheme}


The hybrid scheme can then be expressed as

a_{E}/D_{e}=\left\{ \begin{matrix}
   -\text{Pe}_{\Delta e}\text{                 Pe}_{\Delta e}<-2  \\
   1-\frac{1}{2}\text{Pe}_{\Delta e}\text{    }-\text{2}\le \text{ Pe}_{\Delta e}\le 2  \\
   0\text{                       Pe}_{\Delta e}>2  \\
\end{matrix} \right.

which can be rewritten in the following compact form

a_{E}/D_{e}=\left[\!\left[ -\text{Pe}_{\Delta e},1-\frac{1}{2}\text{Pe}_{\Delta e},0 \right]\!\right]


The coefficient for the west neighbor grid point can be obtained using a similar approach.

a_{W}/D_{w}=\left[\!\left[ \text{Pe}_{\Delta w},1+\frac{1}{2}\text{Pe}_{\Delta w},0 \right]\!\right]


The above hybrid scheme combines the advantages of the central difference and upwind schemes to yield better results for cases where \left| \text{Pe}_{\Delta } \right|\to \infty or \left| \text{Pe}_{\Delta } \right|\sim 0. However, there is still room for improvement of the solution when \left| \text{Pe}_{\Delta } \right| is near 2 (see Problem 4.23).