A New Explicit Numerical Schemes for Time-Dependent PDEs with Application to Pressure Driven Fluid Flow in a Rectangular Duct

Abstract

A modified class of temporal discretization schemes for partial differential equations (PDEs) is proposed, explicit and second to fifth-order accurate in time. In time, the stability region of the proposed modified second-order scheme is larger than the standard second-order Adams–Bashforth method constructed on two time levels. A modification made for the Du Fort–Frankel method was also implemented in the proposed second-order scheme, which permits the little larger stability region, but the scheme becomes first-order accurate. Since the Du Fort–Frankel method cannot be employed without a modification of averaging in time levels, the proposed second-order scheme can be used without any modification. The proposed modified scheme with different orders in space and second orders in time was implemented for heat and mass transfer of chemically reactive fluid flow in a rectangular duct. The flow is generated due to applying different pressure gradients. The contour plots of velocity, temperature, and concentration profiles are portrayed at different pressure gradients; Péclet number in heat transfer, Péclet number in mass transfer, reaction parameter, and at different times. In addition, stability and convergence conditions for the considered system of linear and non-linear PDEs consisting of non-dimensional momentum, energy, and concentration equations were found for two cases. The displayed graphs depict the transfer of heat in the fluid, which rises due to heated boundaries, and the transfer of mass in the fluid at various moments. Classical models can be solved using the proposed method, which has a faster convergence rate than the standard or classical approach. This approach is illustrated through computer simulations that demonstrate its key computational features. It is believed that the data presented in this study will serve as a useful source for future fluid flow investigations to be conducted in an industrial setting within an enclosed area.