A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection

We construct a second-order, linear, and fully decoupled scheme for Rayleigh–Bénard convection between Navier-slip walls. The nonlinear terms are treated explicitly and multiplied by a scalar with exact value $1$ and no energy contributed to the continuous problem. As a result, each BDF2 step, with a rotational pressure correction, solves only constant-coefficient linear problems. For the time discretization, we prove that, for any time-step, the energy decays when the walls have equal temperatures, under a condition on the thermal weight, and the energy is bounded on finite time intervals when heated from below. Relaxing the scalar towards $1$ preserves both estimates. Energy stability does not control the accuracy of the scalar; the velocity and temperature fields are computed with its unrelaxed value, whose deviation from $1$ scales like $(E\mathrm{/}C)(\delta t\mathrm{/}\tau {)}^3$ for kinetic energy $E$, time-scale $\tau$ of the flow, and weight $C$ of the scalar equation. In our tests, choosing $C$ comparable to $E$ keeps this deviation small and doubles the largest time-step for which steady rolls are reproduced to a relative difference of ${10}^{-8}$. With Taylor–Hood elements, we observe second-order convergence in time, recover the critical Rayleigh numbers, and match published Nusselt numbers to within $0.5\mathrm{\%}$. A three-dimensional implementation passes corresponding validation tests.