A Stable Semi-Implicit Two-Step GFDM for the k–ω Turbulence Model: Positivity, Boundedness, and Energy Stability on Non-Uniform Grids

Abstract

A modification in explicit, unconditionally stable scheme is proposed for solving timedependent partial differential equations. In contrast to the conventional DuFort–Frankel method, the proposed scheme allows sufficient positivity conditions to be identified for the resulting numerical solution. The proposed scheme is first-order accurate in time, while its spatial accuracy depends on the order of the generalized finite difference method (GFDM) employed. Fourier series analysis is provided for proving the scheme to be unconditionally stable. The plots for stability regions of the scheme are also provided. The convergence of the scheme is also provided for system of convection diffusion equations. The scheme is applied for solving k – ω turbulent model of non-Newtonian fluid flow over Riga plate with heat and mass transfer. An energy estimate of proposed scheme for nonlinear model of Turbulent flow under the effects of heat and mass transfer on non-uniform grid is also established. A non-uniform grid is constructed for considered turbulent boundary layer flow in which fine grid is used near the wall/plate and course grid is used away from the plate. Furthermore, positivity and boundedness of proposed scheme are established for Stokes’ first problem on non-uniform grid. The comparison of proposed scheme is made with existing Euler and DuFort Frankel methods and it provides less error than existing scheme on most of chosen step sizes.