Solve the 1D heat equation on a grid

Approximate one-dimensional heat diffusion from a centred grid impulse with fixed-zero endpoints.

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Tool1D Heat Equation Solver
Input
Output
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Preview

How it works

Each interior value advances by the forward-time centred-space difference uᵢⁿ⁺¹ = uᵢⁿ + r(uᵢ₋₁ⁿ − 2uᵢⁿ + uᵢ₊₁ⁿ). The time-step count keeps r = diffusivity × Δt ÷ Δx² at or below 0.45 while endpoint values stay zero.

Formula
r = diffusivity × Δt ÷ Δx² ≤ 0.45
  • The 0.45 ceiling sits below the explicit one-dimensional stability bound of 0.5.
  • Length, grid points, diffusivity, elapsed time, and peak remain physical modelling inputs rather than defaults.

Worked example

Solve the 1D heat equation on a grid
Input
											Length: 1
Points: 11
Diffusivity: 0.01
Seconds: 1
Initial peak: 100
Output
												X    Value
---  ---------
0    0
0.1  0
0.2  3.703704
0.3  11.111111
0.4  22.222222
0.5  25.925926
0.6  22.222222
0.7  11.111111
0.8  3.703704
0.9  0
1    0

When to use this

Numerical-method lessons demonstrate diffusion, convergence exercises refine the spatial grid, and heat prototypes export samples for a plot.

Edge cases

  • Changing grid density changes the physical width and total quantity represented by the one-node initial peak.
  • Fixed-zero endpoints model permanent sinks rather than insulated boundaries.
  • Stability prevents blow-up but does not prove spatial accuracy or agreement with an analytic solution.