Classical Electrodynamics · Problem 1.24

Boundary-Value Problems in Electrostatics: I — John David Jackson

Physics — E&M · relaxation-method-poisson-unit-square

Problem

Consider solution of the two-dimensional Poisson equation problem of Problem 1.21, a unit square with zero potential on the boundary and a constant unit charge density in the interior, by the technique of relaxation. Choose h=0.25h = 0.25 so that there are nine interior sites. Use symmetry to reduce the number of needed sites to three, at (0.25,0.25)(0.25, 0.25), (0.5,0.25)(0.5, 0.25), and (0.5,0.5)(0.5, 0.5). With so few sites, it is easy to do the iterations with a block of paper and a pocket calculator, but suit yourself. ((a)) Use the ``improved grid'' averaging of Problem 1.22 and the simple (Jacobian) iteration scheme, starting with 4πϵ0Φ=1.04\pi\epsilon_0\Phi = 1.0 at all three interior sites. Do at least six iterations, preferably eight or ten. ((b)) Repeat the iteration procedure with the same starting values, but using Gauss--Seidel iteration. ((c)) Graph the two sets of results of each iteration versus iteration number and compare with the exact values, 4πϵ0Φ(0.25,0.25)=0.56914\pi\epsilon_0\Phi(0.25, 0.25) = 0.5691, 4πϵ0Φ(0.5,0.25)=0.72054\pi\epsilon_0\Phi(0.5, 0.25) = 0.7205, 4πϵ0Φ(0.5,0.5)=0.92584\pi\epsilon_0\Phi(0.5, 0.5) = 0.9258. Comment on rate of convergence and final accuracy.
Intuitive explanation

The relaxation method solves Laplace's equation iteratively by updating each grid point based on its neighbors' values, using the discrete Laplacian approximation. The key insight is that Jacobian iteration (updating all points simultaneously from old values) converges more slowly than Gauss–Seidel iteration (updating each point using the newest available values), and symmetry lets you compute just three sites instead of nine. Watch how quickly each method approaches the exact solution and why one outpaces the other.

Intuitive explanation

The relaxation method solves Laplace's equation iteratively by updating each grid point based on its neighbors' values, using the discrete Laplacian approximation. The key insight is that Jacobian iteration (updating all points simultaneously from old values) converges more slowly than Gauss–Seidel iteration (updating each point using the newest available values), and symmetry lets you compute just three sites instead of nine. Watch how quickly each method approaches the exact solution and why one outpaces the other.

Related theory

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