**Homework 3: 2-D Convection (Computational Programming)**

**Problem 1 (100 points)**

Consider two-dimensional steady-state convection without diffusion (Γ = 0) over a square domain define by and governed by the partial differential equation (PDE):

and with the boundary conditions such that along the left edge (*x* = 0) , along the bottom edge (*y* = 0) , and along the other edges, the gradient of in the stream-wise direction is zero. The velocity field is defined as .

*Problem setup and analytical solution:*

Problem setup |
Contour plot of from analytical solution |
Line plot of at X–X axis from analytical solution |

__Find:__

the discretization equations for the above governing PDE by using__Derive__**(a)**first-order up-winding method, and**(b)**central differencing method.by using__Contour plots of__**(a)**first-order up-winding method, and**(b)**deferred-correction method.across the domain (along the__Line plots of__*X*–*X*axis) by using**(a)**first-order up-winding method, and**(b)**deferred-correction method.the numerical simulation results by thefirst-order up-winding method andthe deferred-correction method with the analytical solution shown in the figuresabove.__Compare__

(*Note:* Blending factor for the deferred correction method.)