[100 points] The two-dimensional Laplace equation around a cylinder with a domain [0, 2π]×[1, 5] is given
by.
∂
2ψ
∂x2
+
∂
2ψ

∂y2

1
r
∂
∂r
r
∂ψ
∂r
+
1
r
2
∂
2ψ
∂θ2
= 0 (1)
The analytical solution is given by
ψ(r, θ) = U∞

r −
R2
r

sin(θ) (2)
(3)
In here R = 1 and U∞ = 1. The Drichlet boundary conditions:
ψ(1, θ) = 0 (4)
ψ(5, θ) =
5 −
1
5

sin(θ) (5)
(6)
Use the second-order accurate finite difference discretization with uniform 81×41, 161×81 and 321×161
meshes to solve the above Laplace equation. For this purpose

1. Implement the fully implicit solution algorithm and use a direct solver (LU factorization) to solve.
2. Implement Jacobi, Gauss-Seidel and SOR algorithms and compare their convergence rates with the
   iteration numbers.
3. Plot Error function versus the mesh space ∆x in a log-log scale. Compute the spatial convergence
   rate.
   The error function is given by
   Error = kψi,j − ψanalytick2
   √
   imaxjmax
   (7)
   [20 points] Why we should not use the following skewed Laplace operator to solve the Laplace equation
   in Cartesian coordinates?
   ∇2u =
   −4ui,j + ui+1,j+1 + ui−1,j−1 + ui−1,j+1 + ui+1,j−1
   2∆x
   2
   = 0

Sample Solution