A Fully Decoupled Gauge–Uzawa Scheme for Thermally Coupled Variable‐Density MHD Equations With Error Estimates

ABSTRACT

We propose a fully decoupled first‐order numerical scheme for the numerical solution of the thermally coupled magnetohydrodynamics (MHD) equations with variable density in this paper. First, we construct an equivalent reformulation of the governing equations. On this basis, a fully decoupled time‐discretization scheme is developed by combining the Gauge‐Uzawa method and the auxiliary variable approach, which decouples velocity and pressure and efficiently handles the nonlinear convective terms. Compared with traditional pressure correction methods, the Gauge‐Uzawa method achieves direct one‐step velocity–pressure decoupling without additional pressure correction steps, thus ensuring efficiency, maintaining pressure accuracy, and improving numerical stability and precision. Second, the unconditional energy stability of the scheme is rigorously proved. Furthermore, we overcome the difficulties caused by the strong nonlinearity brought by variable density and the strong coupling among thermal‐fluid‐magnetic multiphysics fields, and complete the first rigorous error analysis of the scheme. Eventually, the validity of the method is confirmed by a sequence of numerical tests.