Numerical Solutions for a PDE using Jacobi, Gauss-Seidel and SOR techniques

Devise appropriate tests of convergence, and thus explore the merits from a computational efficiency viewpoint the Jacobi, Gauss-Seidel and Successive Over-relaxation (SOR) techniques to solve the PDE for τ = 0.05. You will need to be able to prescribe appropriate criteria of where you place your far-field domains in x = ±∞ and y∞, and prescription of number of unknowns (i.e (L, N) grid points) as well as work to the limitations of your computer/laptop! Describe in your reporting the convergence criteria you used to confirm grid independence of your numerical solutions. A convenient property of interest is the surface velocity perturbation component Usurf = u = ∂φ ∂x on the aerofoil surface x(0 : 1), approximated here to be at y = 0, which you should use to ascertain what 2 your solution looks like. Make plots of how Usurf varies with x(0 : 1) and along the entire y = 0, x(−q : s) plane. Likewise make plots of the surface v− velocity component along y = 0. A colour plot of your entire (u, v) perturbation field solution over your complete (x, y) computational domain will also be instructive – use the pcolor matlab command, or alternative. 

For the SOR method make a plot of convergence acceleration achieved as you fix the number of L×N unknowns with the acceleration parameter ω. Further make plots of optimal ω giving you the fastest convergence as you vary the grid resolution through varying L, N. Compare your findings with the theoretically expected rate/curves and comment.

By a coordinate transformation, namely: η = y − yb(x), 0 ≤ x ≤ 1, (7) transform Eqn. 1 and thus the flow tangency boundary condition to the new (x, η)-coordinate plane. Hence modify the SOR code developed earlier to solve your transformed equations. Outline clearly in your report the form of the equations that you discretise. Compare your result for τ = 0.05 obtained with the transformed equation set with that found earlier. Subsequently investigate how/whether your solutions differ, as you increase/decrease τ the aerofoil thickness and with solutions obtained with the untransformed equation. Comment on any special features in the solution or tricky numerical issues that you had to circumvent to compute the most accurate solutions.

For the Gauss-Seidel method, develop a simple multigrid approach to accelerate convergence of your method. Compare the numerical efficiency (computer timing) and convergence acceleration that arises with your multigrid approach. Choose appropriate and reasonable number of (L, N) grid points that you are able to compute solutions for in good time with your computer. Your reporting should highlight aspects of convergence efficiencies and speedup of your code achieved, as you vary (L, N) with the multigrid approach.