🌡️ Heat Equation — 2D Temperature Distribution

Draw heat sources and cold sinks on a 2D plate, then watch the temperature field evolve. Toggle between transient (time-stepping) and steady-state (iterative Gauss-Seidel) modes to explore how heat conducts through materials.

Drawing Tool

Simulation

Diffusivity α 0.20 Brush size 5 px

Temperature (°C)

Max—

Min—

Mean—

Step0

0°50°100°

∂T/∂t = α·∇²T
FDM: T[i][j]_n+1 = T_n +
α·Δt/Δx²·(T_left+T_right+
T_up+T_down−4·T)
Stability: α·Δt/Δx² ≤ 0.25

Heat Conduction Physics

The heat equation ∂T/∂t = α∇²T describes how temperature diffuses in a medium, where α = k/(ρcₚ) is the thermal diffusivity (thermal conductivity divided by density × specific heat). The finite difference method (FDM) approximates derivatives on a discrete grid: the Laplacian ∇²T ≈ (T_i+1,j + T_i-1,j + T_i,j+1 + T_i,j-1 − 4T_i,j) / Δx². For explicit time-stepping, the stability condition α·Δt/Δx² ≤ 0.25 must be satisfied to prevent numerical blow-up. The steady-state (∂T/∂t = 0, Laplace equation ∇²T = 0) can be found iteratively using Gauss-Seidel relaxation.