Low-cost adaptive pressure-correction projection method for the 2D/3D time-dependent natural convection problems

The paper firstly develops a high-order and low-complexity time-stepping technique for solving two- and three-dimensional (2D/3D) natural convection problems, which is based on the standard pressure-correction projection (PCP) method and achieves higher-order accuracy by introducing the time filter (TF) technique. The new method can offset the weakness of PCP method while preserving its inherent advantages and can achieve higher-order in time by making a minimally intrusive modification to the PCP program at no extra computational and cognitive complexity. More importantly, it generates a low cost error estimator for adapting the time stepsize that can enhance time efficiency and reliability. Subsequently, the unconditional stability and error analysis of the fully-discrete PCP plus TF (PCP+TF) finite element method are proved. Notably, we construct low-complexity adaptive PCP algorithm, adaptive PCP+TF algorithm and the variable step, variable order (VSVO) algorithm using the TF technique. Ultimately, we verify the above viewpoints and theoretical analysis through some 2D/3D numerical experiments.