This paper is devoted to the study of numerical methods used in the analysis of a thermal transient flight cycle. Two priorities are stressed. The first one concerns a stability issue, namely the fluid-solid interface conditions. The main properties of the Dirichlet-Robin and Neumann-Robin conditions are first recalled, before proposing for the first time an adaptive heuristic condition combining them. The second line of research is dedicated to an accuracy issue, the temporal interpolation of the fluid conditions for dealing with non-coupling instants. We propose here to adopt a physics-based approach by naturally considering the flux-temperature linearity. Then, using a simple numerical test under severe thermal boundary conditions, three test cases are proposed. The first one demonstrates that the adaptive interface condition always produces smooth solutions. The other two cases show that the physical interpolation approach is much more accurate than the one based on temporal linearity, using a costly but accurate reference solution built for the validation. Moreover, important improvements were made by adequately defining the integral contribution of the boundary condition using the Gauss points in the finite element solid code. It was also revealed that a relevant approximation of the heat transfer requires a sufficiently fine solid mesh at the interface. Finally, potential improvements are proposed at the end of this paper and more particularly two other interface conditions. One of them has the remarkable property of not using the convective heat transfer coefficient.