In order to examine the numerical instability in 1D non-local turbulent theory, an attempt was made to solve a simplified version of Xiong's non-local turbulent model taking into account the equation of conservation of energy. Using a finite-difference method and staggered grid, the structures of three main configurations of stellar convective zones were calculated. The numerical results show that the convective flux becomes negative and the radiative flux exhibits bumps across the boundaries of the convectively unstable zone. The turbulent velocity decreases rapidly in the upward direction when the pressure decreases to a certain level. In A–F type stars, the convective zones are separated when the temperature gradient is taken as a proxy, but merge when the turbulent velocity is taken as a proxy. The anisotropic treatment of turbulent convection facilitates the non-local effect at the upper boundary of the convection zone.
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Tao Cai (2014) studied this question.