Numerical Investigation of the 1D Transient Heat Conduction Equation

Comparative finite-difference study of the transient 1D heat equation using Python, NumPy and Matplotlib.

The repository investigates numerical stability, transient diffusion behavior and solution accuracy across multiple finite-difference schemes through comparison with the analytical solution.

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Governing Equation

∂T/∂t = α ∂²T/∂x²

where:

• T(x,t) → temperature distribution
• α = 0.02 m²/hr → thermal diffusivity

Initial condition:

• triangular temperature profile
• T = 0 K at x = 0 and x = 1 m
• T = 100 K at x = 0.5 m

Boundary condition:

• T(0,t) = T(1,t) = 0 K

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Numerical Methods

The following finite-difference schemes are implemented and compared:

• Explicit FTCS

  • forward-time centered-space formulation
  • conditionally stable

• Implicit Laasonen

  • implicit tridiagonal formulation
  • unconditionally stable

• Crank–Nicolson

  • second-order implicit scheme
  • improved transient accuracy

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Validation & Investigation

All numerical schemes are benchmarked against the analytical solution for multiple test cases involving varying:

• mesh sizes
• timestep values
• Fourier number conditions

Current investigations include:

• timestep sensitivity
• numerical stability behavior
• transient diffusion response
• explicit scheme instability
• comparative scheme accuracy

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Repository Structure

project/
│
├── core/
│   ├── analytical_solution.py
│   └── intial_comditions.py
│
├── plots/
│   ├── test_case_1.png
│   ├── test_case_2.png
│   ├── test_case_3.png
│   ├── test_case_4.png
│   ├── test_case_5.png
│   ├── test_case_6.1.png
│   ├── test_case_6.2.png
│   └── test_case_6.png
│
├── solvers/
│   ├── crank_nicolson.py
│   ├── explicit_ftcs.py
│   └── implicit_laasonen.py
│
├── studies/
│   ├── case_1.py
│   ├── case_2.py
│   ├── case_3.py
│   ├── case_4.py
│   └── case_5&6.py
│   
├── LICENSE
└──README.md

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Current Limitations

• restricted to one-dimensional conduction
• constant thermal diffusivity assumption
• no multidimensional extension
• no adaptive meshing
• convergence study not yet implemented

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Engineering Relevance

This study investigates foundational numerical behaviors that appear throughout computational heat transfer and broader PDE-based simulation methods.

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Tools Used

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