For the finite difference scheme in Problem 4:

a) Write the formulas for the Gauss-Seidel algorithm in terms of the grid point values.

b) Prove convergence of the Gauss-Seidel iteration procedure.

Problem 4

The two-dimensional heat conduction equation (8.15) is solved in the rectangular domain \(0 \leq x \leq L_{x}, 0 \leq y \leq L_{y}\) with the boundary conditions \(u=0\) at \(x=0, x=L_{x}, y=0, y=L_{y}\). The solution uses the Crank-Nicolson central difference scheme on a structured, uniform, Cartesian grid (see (8.17) and (8.18)). Rewrite the problem in matrix form. Develop the row equation and expressions for the coefficients as in (8.9) and (8.10). Include the rows corresponding to the boundary conditions.