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AMATH 581 ( c○J. N. Kutz) 48k x k y k z k0 1.0 2.0 2.01 1.75 3.75 2.952 1.95 3.96875 2.98625. . . .10 2.0 4.0 3.0Table 7: Convergence of Gauss-Seidel iteration scheme to thesolutionvalueof(x, y, z) =(2, 4, 3) from the initial guess (x 0 ,y 0 ,z 0 )=(1, 2, 2).scheme which can be implemented in an effort to enhance convergence. It isalso possible to use several previous iterations to achieve convergence. Krylovspace methods [6] are often high end iterative techniques especially developedfor rapid convergence. Included in these iteration schemes are conjugant gradientmethods and generalized minimum residual methods which we will discussand implement [6].Application to Advection-DiffusionWhen discretizing many systems of interest, such as the advection-diffusionproblem, we are left with a system of equations that is naturally geared towarditerative methods. Discretization of the stream function previously yielded thesystem−4ψ mn + ψ (m+1)n + ψ (m−1)n + ψ m(n+1) + ψ m(n−1) = δ 2 ω mn . (2.3.10)The matrix A in this case is represented by the left hand side of the equation.Letting ψ mn be the diagonal term, the iteration procedure yieldsψ k+1mn= ψk (m+1)n + ψk (m−1)n + ψk m(n+1) + ψk m(n−1) − δ2 ω mn4. (2.3.11)Note that the diagonal term had a coefficient of |−4| =4andthesumoftheoffdiagonal elements is |1| + |1| + |1| + |1| =4. Thusthesystemisattheborderlineof being diagonally dominant. So although convergence is not guaranteed,itishighly likely that we could get the Jacobi scheme to converge.Finally, we consider the operation count associated with the iterationmethods.This will allow us to compare this solution technique with Gaussian eliminationand LU decomposition. The following basic algorithmic steps are involved:1. Update each ψ mn which costs N operations times the number of non-zerodiagonals D.