Theoretical study of the stability in transient neutronics/thermal-hydraulics coupling problems using Fourier analysis

This paper theoretically studies the stability of Picard iteration in transient neutronics and thermal-hydraulics coupling problems using Fourier analysis. Various iteration schemes of a simplified transient coupling problem were mathematically described in one-dimensional Cartesian coordinates. Fourier analysis was then applied to the iteration schemes and expressions of iterative spectral radius were obtained. The results show that the spectral radius of the deterministic iteration is identical to that of a particular case of the Monte Carlo-based iteration, indicating their mathematical equivalence. Verifications confirm the validity of the methodology and show that the theoretical model agrees well with numerical results. Analytical results show that the iterative stability is primarily affected by the system size, feedback strength, scattering ratio, and time step. The stability is improved for systems with small sizes, and unconditional stability is achieved for systems with relatively weak feedback. The time step effect is highly coupled with multiple factors including the system sizes, initial state, and reactivity. The selection of the time step becomes more stringent as the reactivity is more positive for supercritical systems, or less negative for subcritical systems. For subcritical systems, reducing the time step does not necessarily improve the stability. Further discussions show the relations of the iterative stability to the thermal-hydraulics properties, under-relaxation treatment, and Monte Carlo cycles, quantitatively showing that the under-relaxation treatment improves the stability, while using excessive Monte Carlo cycles worsens the stability and may even lead to severe divergence.