2D Transient Heat Equation Simulator — Time-Evolving Diffusion Visualizer

The core physics is described by Fourier's Law of Heat Conduction, leading to the 2D Transient Heat Equation. It states that the rate of temperature change at a point is proportional to how much the temperature curves around that point.

$T(x,y,t)$ is the temperature field (°C). $\alpha$ is the thermal diffusivity (m²/s). The terms $\frac{\partial^2 T}{\partial x^2}$ and $\frac{\partial^2 T}{\partial y^2}$ are the spatial curvatures of temperature, which drive the heat flow.

To solve this on a computer, we discretize it using the Explicit Finite Difference Method (FDM). This equation is what the simulator calculates for every grid point at every time step to create the animation.

Here, $T_{i,j}^n$ is temperature at grid point (i,j) at time step n . $r = \alpha \Delta t / (\Delta x)^2$ is a dimensionless stability parameter. The term in parentheses sums the temperatures of the four neighboring cells, minus four times the center cell-this is the discrete version of the curvature ($\nabla^2 T$).

Electronics Cooling (PCB & Chip Design): Engineers use this exact analysis to prevent overheating. By simulating heat diffusion from hot components like CPUs across a circuit board, they can design optimal heat sink placement and airflow, ensuring reliability. The "add custom heat sources" feature in the simulator mimics placing hot components on a board.

Building Science & Insulation: How does heat leak through a wall or window over a day-night cycle? Transient 2D analysis models the thermal mass and insulation properties of building materials, helping architects design energy-efficient homes that stay warm in winter and cool in summer.

Welding & Metal Processing: The intense heat of a weld creates a complex, moving temperature field in the metal. Simulating this "heat-affected zone" is crucial to predict residual stresses, distortions, and material property changes, ensuring the weld's structural integrity.

Geothermal Systems & Ground-Source Heat Pumps: To design efficient systems, engineers model how heat injected into or extracted from the ground propagates through the soil over months and years. This 2D transient analysis helps size the system and predict long-term performance.

First, do not confuse "simulation speed" with "the rate of physical time progression". Speeding up the animation with the slider does not change how fast heat actually propagates; it is merely the playback speed for visualization. The progression of physical time is determined by the "time step Δt" and the "number of calculation steps". For example, calculating 1000 steps with Δt=0.1 seconds means you are observing the state after 100 seconds.

Next, "thermal diffusivity α" and "thermal conductivity λ" are different. α represents "how easily heat spreads", while λ represents "how easily heat is transferred". Their relationship is given by $\alpha = \lambda / (\rho c)$, involving density ρ and specific heat c. Insulating materials have a small λ, but they may also have small ρ and c, sometimes resulting in a surprisingly large α. When considering a material's "thermal character", get into the habit of thinking about these three property values (λ, ρ, c) as a set.

Finally, the "divergence" in this tool is not a real explosion. When the Courant number r exceeds 0.25, the calculation breaks down, but this is a limitation of the numerical method (explicit method). In practice, refining the mesh requires an extremely small Δt, leading to enormous computation times. This is precisely why other stable algorithms, such as implicit methods or semi-implicit methods, have been developed.