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This paper considers a Leray regularization model of incompressible,non-isothermal fluid flows which uses nonlinear filtering based on indicatorfunctions, and introduces an efficient numerical method for solving it. Theproposed method uses a multi-step, second-order temporal discretization with afinite element (FE) spatial discretization in such a way that the resultingalgorithm is linear at each time level, and decouples the evolution equationsfrom the velocity filter step. Since the indicator function chosen in thismodel is mathematically based on approximation theory, the proposed numericalalgorithm can be analyzed robustly, i.e the stability and convergence of themethod is provable. A series of numerical tests are carried out to verify thetheoretical convergence rates, and to compare the algorithm with directnumerical simulation and the usual Leray-$\alpha$ model of the flow problem.

Improving accuracy in the Leray model for incompressible non-isothermal flows via adaptive deconvolution-based nonlinear filtering