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• Four integrated SPH models are presented and compared for simulating natural convection. • The most suitable SPH model for natural convection is provided. • Natural convection for R a = 10 9 and Pr = 0.71 is successfully modeled for the first time by using the SPH method. • Mechanisms of the natural convection in a square cavity at different Rayleigh numbers are discussed. This paper conducts the simulation of natural convection in a differentially heated square cavity at high Rayleigh numbers by using the smoothed particle hydrodynamics (SPH) method. Due to the decrease of the accuracy and stability, it is challenging for the SPH method to simulate natural convection at high Rayleigh numbers, and there are few reported SPH literatures of natural convection at R a > 10 6 for air (Pr = 0.71). In this study, four integrated SPH models are presented to simulate the natural convection and their accuracy and stability are assessed. These four SPH models are associated with Kernel Gradient Correction (KGC) to improve approximation accuracy and Particle Shifting Technology (PST) to regularize particle distribution while they are different in treating density diffusion and calculating the pressure term. The numerical results show that SPH model_4 (KGC, PST, δ -SPH and asymmetric pressure approximation) is the most suitable for simulating the closed natural convection problems, especially at high Rayleigh numbers. Good agreements with reference solutions are obtained by SPH model_4 for the natural convection at 10 4 ≤ R a ≤ 10 8. Furthermore, the simulation of natural convection at R a = 10 9 is conducted by SPH model_4. The evolutions of thermal convection are described in detail. It is found that dynamics characteristic reveals that the dominant force is the pressure gradient, rather than the buoyancy force before the quasi-steady state. In addition, the chaotic motion at R a = 10 9 has significant influence to the heat transfer characteristic in the vertical boundary layers. [ABSTRACT FROM AUTHOR]