A finite volume scheme is developed using a two-dimensional structured Cartesian grid with constant steps \(\Delta x\) and \(\Delta y\) (see Figure 5.5a). Write the following approximations using the values of the function \(u\) at the grid points, such as \(\mathrm{P}, \mathrm{E}, \mathrm{EE}\), and W.

a) Approximation based on linear interpolation of the convective flux integral \(\int_{S_{\mathrm{c}}} u \boldsymbol{V} \cdot \boldsymbol{n} d S\), where \(S_{\mathrm{e}}\) is the face containing the point

e, \(\boldsymbol{V}\) is the constant velocity \(\boldsymbol{V}=(1,0.5)\), and \(\boldsymbol{n}\) is the outward-facing unit-length normal to \(S_{\mathrm{e}}\).

b) The same as in

(a) but with \(\boldsymbol{V}=(0.0,2.5)\).

c) The same as in

(a) but the approximation must be based on the QUICK interpolation.

d) Second-order approximation of the diffusive flux integral \(\int_{S_{\mathrm{e}}} \chi(u) abla u \cdot \boldsymbol{n} d S\).