An efficient semi‐implicit temporal scheme for boundary‐layer vertical diffusion

Time integration of the boundary‐layer vertical diffusion equation has been investigated. The nonlinearity associated with the diffusion coefficient makes the implicit approach impractical, while the use of an explicit scheme limits the stable time‐step sizes and consequently would be inefficient. By using a diagonally implicit Runge–Kutta scheme, a new approach has been proposed in which the diffusion coefficients at each internal stage are calculated by a weight‐averaged combination of solutions. Using the weight coefficient α offers more robust calculations due to involving implicit solutions and, as shown, it could improve the accuracy due to more engaging the explicit solutions. It has been found that the proposed semi‐implicit method is more accurate and computationally less expensive than the implicit scheme. Moreover, in terms of stability and accuracy improvement, the advantage of the proposed DIRK scheme, compared to the scheme proposed by Diamantakis et al. (), has been revealed, particularly for a highly nonlinear diffusion term. The nonlinear vertical diffusion turbulent boundary‐layer flows can be solved more efficiently by the weight‐averaged semi‐implicit Runge–Kutta scheme. The accuracy is improved and the calculation cost is competitive to the current semi‐implicit schemes.