Modified class of explicit and enhanced stability region schemes: Application to mixed convection flow in a square cavity with a convective wall. (30th December 2020) - Nawaz, Yasir; Arif, Muhammad Shoaib
Abstract: A modification of Adams–Bashforth methods is given to construct time discretization schemes for partial differential equations. The second‐order modified method is shown to have a larger stability region than second‐order standard Adams–Bashforth for the two‐dimensional heat equation. Later the scheme is applied on considered flow problem in a square cavity. The flow problem is a modified mathematical model of the heat and mass transfer of mixed convection flow in a square cavity with effects of the inclined magnetic field and thermal radiations. In addition to this, another feature of the present contribution is to apply the coupling approach for employing a mixture of stable and unstable schemes. This coupling approach is based upon the difference quotient that has been used in the literature to construct flux limiters for reducing oscillations in the discontinuous solutions of hyperbolic conservation laws. Since proposed scheme produces oscillation in the beginning and then diverges for the chosen diffusion number that falls in the unstable region, so these oscillations, due to instability, is reduced by coupling it with the scheme that can produce the convergent solution. The convergence of the proposed scheme for the considered modified nondimensional mathematical model of mixed convection flow is also given. The improvement is shown in graphs when proposed second order in time scheme is compared with the standard second order in time Adams–Bashforth method. Also, the mixture of first‐order and unconditionally unstable Richardson's schemes is applied, and the solution is obtained, and some plots are provided. Abstract : This contribution comprises a proposed class of explicit time discretization schemes for time‐dependent partial differential equations. Application of the proposed scheme is given for the transfer of heat to a mixed convective fluid flow in a square cavity. The proposed second‐order scheme has been shown to give enhanced stability region than the existing second‐order scheme. The procedure for coupling schemes is also given that the stability regions of the schemes can be enlarged, but the order of the scheme may be reduced.