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 \(\mathrm{W}\).

a) Approximation of the second order for the volume integral \(\int_{\Omega_{i}} u d \Omega\).

b) Upwind approximation for the convective flux integral \(\int_{S_{\mathrm{e}}} u \boldsymbol{V}\). \(\boldsymbol{n} d S\), 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 \(S_{\mathrm{e}}\).

c) The same as in

(b) but with velocity \(\boldsymbol{V}=(-0.5,1)\).

d) Central difference approximation of the second order for the diffusive flux integral \(\int_{S_{\mathrm{e}}}(\partial u / \partial x) d S\).