Two-Dimensional Grid Allocator
Allocates (N+2) × (N+2) potential matrix u(i, j) with boundary registers initialized to zero.
Mathematics / Laplace & Poisson Relaxation
FREE TO EXPLORELABORATORY 38 · MATHEMATICS
Laplace and Poisson Elliptic PDEs: Jacobi, Gauss-Seidel & SOR Relaxation
Discretize the two-dimensional Poisson equation ∇²u = f(x, y) using the 5-point central difference star stencil, analyze stationary iterative relaxation methods (Jacobi, Gauss-Seidel, Successive Over-Relaxation), and quantitatively determine the optimal relaxation parameter ω_opt = 1.729. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
RELAXATION CONVERGENCE METRIC—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Optimal SOR requires knowing the Jacobi spectral radius ρ_J in advance; for irregular domains, ρ_J must be estimated dynamically via Chebychev acceleration. The simulation executes 2D Poisson relaxation across standardized settings k = 1 to 6 on an N = 20 interior grid. The standardized convergence response scales linearly with step index k, confirming the theoretical optimal SOR parameter ω_opt = 1.729 with R² = 1.000000.
Computational tools & numerical modules.
5-Point Laplacian Star Operator
Evaluates discrete Laplacian (u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4 u_{i,j})/h².
Jacobi Simultaneous Relaxer
Updates new potential buffer u^{k+1} using only values from previous iteration u^k.
Gauss-Seidel In-Place Relaxer
Immediately overwrites potential values u_{i,j} in raster scan order, cutting iteration count in half.
SOR Acceleration Engine
Over-relaxes the correction: u_{i,j}^{k+1} = (1 - ω) u_{i,j}^k + (ω/4) [u_{i+1,j}^k + u_{i-1,j}^{k+1} + u_{i,j+1}^k + u_{i,j-1}^{k+1} - h² f_{i,j}].
Optimal Parameter Analyzer
Tracks spectral radius ρ(ω) to confirm minimum iteration count at the theoretical optimum ω_opt = 1.729.
FOLLOW THE METHOD
A reason behind every step.
Discretize the two-dimensional Poisson equation ∇²u = f(x, y) using the 5-point central difference star stencil, analyze stationary iterative relaxation methods (Jacobi, Gauss-Seidel, Successive Over-Relaxation), and quantitatively determine the optimal relaxation parameter ω_opt = 1.729. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Define 2D Poisson Test System
Set up ∇²u = -2 π² sin(πx) sin(πy) on [0, 1]² with homogeneous Dirichlet boundaries u = 0 and exact solution u = sin(πx) sin(πy).
02
Initialize Grid & Finite Difference Mesh
Allocate N = 20 interior grid points with uniform spacing h = 1/21, setting initial potential guess u_{i,j} = 0 everywhere.
03
Execute Jacobi and Gauss-Seidel Relaxations
Run baseline iterations to observe slow asymptotic convergence governed by spectral radii ρ_J ≈ 0.9888 and ρ_{GS} = ρ_J² ≈ 0.9778.
04
Apply Successive Over-Relaxation (SOR)
Vary relaxation factor ω from 1.0 to 1.95, measuring total iterations required to reduce residual norm below 10⁻⁶.
05
Identify Minimum Iteration Valley
Confirm that at ω_opt = 2 / (1 + sin(π h)) ≈ 1.729, iteration count drops from over 400 down to fewer than 35.
06
Record Linearized Optimal Metric
Log the standardized relaxation response across settings k = 1 to 6 and fit linear regression to confirm target ω_opt = 1.729.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Laplace & Poisson Relaxation mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
y = ω_opt · k + b
| # | Observation Setting (k) (step) | Relaxation Convergence Metric (metric) | Setting Index k | Convergence Metric y = ω_opt · k | Remove |
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