Question 12.1.2: Use the Poisson finite-difference method with n = 6 , m = 5,...
Use the Poisson finite-difference method with n = 6 , m = 5, and a tolerance of 10^{-10} to approximate the solution to
\frac{\partial^{2} u}{\partial x^{2}}(x, y)+\frac{\partial^{2} u}{\partial y^{2}}(x, y)=x e^{y}, \quad 0<x<2, \quad 0<y<1 ,
with the boundary conditions
\begin{array}{ll} u(0, y)=0, & u(2, y)=2 e^{y}, \quad 0 \leq y \leq 1 \\ u(x, 0)=x, & u(x, 1)=e x, \quad 0 \leq x \leq 2 \end{array}
and compare the results with the exact solution u(x, y)=x e^{y} .
Using Algorithm 12.1 with a maximum number of iterations set at N = 100 gives the results in Table 12.2. The stopping criterion for the Gauss-Seidel method in Step 17 requires that
\left|w_{i j}^{(l)}-w_{i j}^{(l-1)}\right| \leq 10^{-10} ,
for each i = 1, … , 5 and j = 1, … , 4. The solution to the difference equation was accurately obtained, and the procedure stopped at l = 61. The results, along with the correct values, are presented in Table 12.2.
Table 12.2
\begin{array}{lcccccc}\hline i & j & x_{i} & y_{j} & w_{i, j}^{(61)} & u\left(x_{i}, y_{j}\right) & \left|u\left(x_{i}, y_{j}\right)-w_{i, j}^{(61)}\right| \\\hline 1 & 1 & 0.3333 & 0.2000 & 0.40726 & 0.40713 & 1.30 \times 10^{-4} \\1 & 2 & 0.3333 & 0.4000 & 0.49748 & 0.49727 & 2.08 \times 10^{-4} \\1 & 3 & 0.3333 & 0.6000 & 0.60760 & 0.60737 & 2.23 \times 10^{-4} \\1 & 4 & 0.3333 & 0.8000 & 0.74201 & 0.74185 & 1.60 \times 10^{-4} \\2 & 1 & 0.6667 & 0.2000 & 0.81452 & 0.81427 & 2.55 \times 10^{-4} \\2 & 2 & 0.6667 & 0.4000 & 0.99496 & 0.99455 & 4.08 \times 10^{-4} \\2 & 3 & 0.6667 & 0.6000 & 1.2152 & 1.2147 & 4.37 \times 10^{-4} \\2 & 4 & 0.6667 & 0.8000 & 1.4840 & 1.4837 & 3.15 \times 10^{-4} \\3 & 1 & 1.0000 & 0.2000 & 1.2218 & 1.2214 & 3.64 \times 10^{-4} \\3 & 2 & 1.0000 & 0.4000 & 1.4924 & 1.4918 & 5.80 \times 10^{-4} \\3 & 3 & 1.0000 & 0.6000 & 1.8227 & 1.8221 & 6.24 \times 10^{-4} \\3 & 4 & 1.0000 & 0.8000 & 2.2260 & 2.2255 & 4.51 \times 10^{-4} \\4 & 1 & 1.3333 & 0.2000 & 1.6290 & 1.6285 & 4.27 \times 10^{-4} \\4 & 2 & 1.3333 & 0.4000 & 1.9898 & 1.9891 & 6.79 \times 10^{-4} \\4 & 3 & 1.3333 & 0.6000 & 2.4302 & 2.4295 & 7.35 \times 10^{-4} \\4 & 4 & 1.3333 & 0.8000 & 2.9679 & 2.9674 & 5.40 \times 10^{-4} \\5 & 1 & 1.6667 & 0.2000 & 2.0360 & 2.0357 & 3.71 \times 10^{-4} \\5 & 2 & 1.6667 & 0.4000 & 2.4870 & 2.4864 & 5.84 \times 10^{-4} \\5 & 3 & 1.6667 & 0.6000 & 3.0375 & 3.0369 & 6.41 \times 10^{-4} \\5 & 4 & 1.6667 & 0.8000 & 3.7097 & 3.7092 & 4.89 \times 10^{-4} \\\hline\end{array}