A two-stage multi-step numerical scheme for mixed convective Williamson nanofluid flow over flat and oscillatory sheets

The novelty of this contribution is to propose an implicit numerical scheme for solving time-dependent boundary layer problems. The scheme is multi-step and consists of two stages. It is third-order accurate in time and constructed on three-time levels. For spatial discretization, a fourth-order compact scheme is adopted. The stability of the proposed scheme is analyzed for scalar linear partial differential equation (PDE) that shows its conditional stability. The convergence of the scheme is also provided for a system of time-dependent parabolic equations. Moreover, a mathematical model for heat and mass transfer of mixed convective Williamson nanofluid flow over flat and oscillatory sheets is modified with the characteristic of the Darcy–Forchheimer model. The results show that the temperature profile rises by developing thermophoresis and Brownian motion parameter values. Also, the proposed scheme is compared with an existing Crank–Nicolson method. It is found that the proposed scheme converges faster than the existing one for solving scalar linear PDE as well as the system of linear and nonlinear parabolic equations, which are dimensionless forms of governing equations of a flow phenomenon. The findings provided in this study can serve as a helpful guide for future investigations into fluid flow in closed-off industrial settings.