Analysis of an Embedded Variable Step Implicit-Explicit Scheme for Natural Convection Problems

Abstract

This report presents a series of implicit-explicit (IMEX) variable stepsize algorithms for natural convection equations. The presented method requires a minimally intrusive modification to an existing program, does not add to the computational complexity, and is conceptually simple. Here, IMEX means the nonlinear term is treated fully explicitly, while the remaining terms are treated implicitly. Due to the increasing demand for low memory solvers, the addition of time adaptive can improve the accuracy and efficiency of the algorithms. For the first-order algorithm, we prove the stability of the variable stepsize backward Euler scheme combined with Adams-Bashforth 2 (VSS BE-AB2) and analyze convergence. Then, the stability of Constant Timestep Filtered-BE-AB2 (BE-AB2+F) is proved. Moreover, we construct adaptive algorithms by extending the approach to variable stepsize. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.