Accuracy and Stability of Finite Difference Schemes for One Dimensional Transient Heat Conduction
Year: 2026
Type: article
Abstract: This study compares explicit Euler, implicit Euler and Crank–Nicolson finite difference schemes for the one dimensional transient heat equation in a homogeneous rod. A centered spatial approximation and a unified θ formulation are used, with tridiagonal systems solved by the Thomas algorithm. Verification is based on an analytical sine solution, coupled space and time refinement, a separate temporal accuracy test, and additional source and boundary checks. At a fixed diffusion number of 0,4, refinement from 10 to 80 spatial intervals reduces the maximum errors from 4,294140×10⁻³ to 6,619528×10⁻⁵ for explicit Euler, from 1,011156×10⁻² to 1,606823×10⁻⁴ for implicit Euler, and from 2,980727×10⁻³ to 4,726121×10⁻⁵ for Crank–Nicolson. Comparison with the exact semidiscrete solution confirms first order temporal accuracy for implicit Euler and second order accuracy for Crank–Nicolson. An intentionally unstable explicit calculation demonstrates the growth of a small high frequency perturbation. The results distinguish stability, accuracy, and damping, and provide a reproducible verification framework rather than experimental validation of a particular material.
Institution: Tashkent State Transport University
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
OpenAlex ID: W7225817533
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