- 2068 Magh · 10 marks
Laplace: five-point formula, 2x2
Short answer
Using the standard five-point formula for Laplace's equation and iterative method (Gauss-Seidel), the interior nodal values are u1 = 1208.3333, u2 = 791.6667, u3 = 1041.6667, u4 = 458.3333.
Given the values of u(x, y) on the boundary of the square as shown in the figure below, evaluate the function u(x, y) satisfying the Laplace ∇²u = 0 at the pivotal points, using standard five point formula iterative method. [Figure: interior points u1 (top-left), u2 (top-right), u3 (bottom-left), u4 (bottom-right). Top boundary: 1000, 1000, 1000 (left to right). Left boundary top to bottom: 1000 (corner), 2000, 2000, 1000 (corner). Right boundary: 500 (next to u2 row), 0 (next to u4 row), 0 (bottom). Bottom boundary: 500, 0.]
Interior points: u₁ (top-left), u₂ (top-right), u₃ (bottom-left), u₄ (bottom-right). The standard five-point formula for ∇²u = 0 is
u(i,j) = ¼[u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1)]
Reading the figure: top 1000, 1000 (above u₁, u₂); left 2000, 2000 (beside u₁, u₃); right 500, 0 (beside u₂, u₄); bottom 500, 0 (below u₃, u₄). Corner values are not used.
Equations
u₁ = ¼(3000 + u₂ + u₃)
u₂ = ¼(1500 + u₁ + u₄)
u₃ = ¼(2500 + u₁ + u₄)
u₄ = ¼(0 + u₂ + u₃)
Iteration (Gauss–Seidel / Liebmann)
Initial value 875 for every unknown (close to the mean of the boundary values); each new value is used at once in the next equation.
| iteration | u₁ | u₂ | u₃ | u₄ |
|---|---|---|---|---|
| 0 | 875.0000 | 875.0000 | 875.0000 | 875.0000 |
| 1 | 1187.5000 | 890.6250 | 1140.6250 | 507.8125 |
| 2 | 1257.8125 | 816.4062 | 1066.4062 | 470.7031 |
| 3 | 1220.7031 | 797.8516 | 1047.8516 | 461.4258 |
| 4 | 1211.4258 | 793.2129 | 1043.2129 | 459.1064 |
| 5 | 1209.1064 | 792.0532 | 1042.0532 | 458.5266 |
| 6 | 1208.5266 | 791.7633 | 1041.7633 | 458.3817 |
| 7 | 1208.3817 | 791.6908 | 1041.6908 | 458.3454 |
| 8 | 1208.3454 | 791.6727 | 1041.6727 | 458.3364 |
| 9 | 1208.3364 | 791.6682 | 1041.6682 | 458.3341 |
| 10 | 1208.3341 | 791.6670 | 1041.6670 | 458.3335 |
| 11 | 1208.3335 | 791.6668 | 1041.6668 | 458.3334 |
| 12 | 1208.3334 | 791.6667 | 1041.6667 | 458.3333 |
| 13 | 1208.3333 | 791.6667 | 1041.6667 | 458.3333 |
Converged to 4 decimals after 13 iterations.
Direct solution of the equations (check)
u₁ = 1208.3333
u₂ = 791.6667
u₃ = 1041.6667
u₄ = 458.3333
Answer: u₁ = 1208.3333, u₂ = 791.6667, u₃ = 1041.6667, u₄ = 458.3333
✓ Answer checked by Claude Opus 5.5 (high effort)
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