Low-cost second-order time-stepping methods for natural convection problem with variable density

This paper develops and analyses new low-cost second-order algorithms for natural convection problems with variable density based on time filter, including constant timestep, variable timestep and adaptive timestep methods. The technique of time filter can improve the time accuracy and enhance the efficiency of the Backward Euler (BE) algorithm. The stability of constant timestep and variable timestep implicit BE schemes are proved and the schemes are unconditionally stable. This is to our knowledge the first provable, while the existing literature focuses on the proof of linear-implicit schemes. We also prove the stability of second-order constant stepsize and variable timestep algorithms and construct adaptive algorithms by extending the approach to variable time stepsize. Finally, numerical examples are implemented to verify the stability and high efficiency of these algorithms.