Comparative analysis of intrusive and non-intrusive methods for computing polynomial chaos expansion coefficients: a case study of the heat equation

This paper considers the heat equation under a formulation that accounts for the stochastic nature of the thermal diffusivity coefficient and randomness in the initial condition. Uncertainty is represented using the Polynomial Chaos Expansion method. The analytical solution is compared with two stochastic approximations: the non-intrusive least-squares method and the intrusive Galerkin method implemented together with the Crank — Nicolson scheme. Statistical simulations are performed, mean temperature trajectories and their three-dimensional visualizations are constructed, and errors between the methods are evaluated. The results demonstrate a high degree of consistency between the numerical approaches and the analytical solution, confirming the practical suitability of the proposed methods.

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