A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection
Abstract
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/C)(δt/τ)3 for kinetic energy E, time-scale τ 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%. A three-dimensional implementation passes corresponding validation tests.
Related
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BibTeX
@misc{mccurdy2026secondorder,
title = {{A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection}},
author = {Matthew McCurdy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31364},
eprint = {2610.31364},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31364},
}
RIS
TY - UNPB
TI - A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection
AU - Matthew McCurdy
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31364
UR - https://arxiv.org/abs/2610.31364
AB - 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/C)(δt/τ)^3 for kinetic energy E, time-scale τ 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%. A three-dimensional implementation passes corresponding validation tests.
ER -