The generalized scalar auxiliary variable applied to the incompressible Boussinesq Equation

Abstract: This paper introduces a second-order time discretization for solving the incompressible Boussinesq equation. It uses the generalized scalar auxiliary variable (GSAV) and a backward differentiation formula (BDF), based on a Taylor expansion around $t^{n+k}$ for $k\geq3$. An exponential time integrator is used for the auxiliary variable to ensure stability independent of the time step size. We give rigorous asymptotic error estimates of the time-stepping scheme, thereby justifying its accuracy and stability. The scheme is reformulated into one amenable to a $H^1$-conforming finite element discretization. Finally, we validate our theoretical results with numerical experiments using a Taylor--Hood-based finite element discretization and show its applicability to large-scale 3-dimensional problems.