Laplace Equation Simulator - Constraint Theory

Iterations: 0 Residual: 0.0000 Status: Ready

Solver Method

Grid Parameters

Grid Size: 64

Max Iterations: 1000

SOR Parameter

Omega (ω): 1.85 Optimal: 1.7-1.9 for most problems

Boundary Value

Value: 1.0

Visualization

Heat Map Equipotential Lines Field Lines (Gradient) Grid Overlay

Animation

Speed: 50 ms/iter

Animate Convergence

Presets

Controls

Convergence

Target: 1e-6

Current: —

Rate: —

Target Residual: 1e-6

About the Laplace Equation

The Laplace equation ∇²φ = 0 describes steady-state systems where the value at any point is the average of its neighbors. This creates a "smooth" field that satisfies boundary constraints everywhere simultaneously.

Physical Examples

Electrostatic potential in charge-free regions
Steady-state heat distribution
Fluid flow in porous media
Gravitational potential in empty space

Numerical Methods

Jacobi: Simple but slow (O(n) iterations)
Gauss-Seidel: Faster, uses updated values
SOR: Fastest with optimal ω (~1.8)

Using the Simulator

Select Boundary Mode and click edges to set fixed values
Use Source/Sink to add interior charges
Choose a Preset for classic configurations
Click Solve to iterate until convergence
Enable Animate to watch convergence in real-time