Spectral Relaxation Method and Spectral Quasilinearization Method for Solving Unsteady Boundary Layer Flow Problems

Abstract

Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box.

Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method

“…The spectral relaxation method (see [39]) was used to solve the system of non-similar equations (8)- (10) with the boundary conditions (11). In the SRM framework, we obtain the iterative scheme…”

The Effects of Thermal Radiation on an Unsteady MHD Axisymmetric Stagnation-Point Flow over a Shrinking Sheet in Presence of Temperature Dependent Thermal Conductivity with Navier Slip

“…In this case the technique is used to linearize and decouple a system of differential equations. Further details of the rules of the SRM can be found in [28, 29]. …”

The Effect of Suction/Injection on Free Convection in a Micropolar Fluid Saturated Porous Medium under Convective Boundary Condition

“…In this system of equations, the nonlinear components can be linearised using one term Taylors series for multiple variables so that the Eqs. (9)- (12) give the following iterative sequence of linear differential equations:…”

“…In this work, as we discussed below, the Chebyshev pseudo-spectral method was used to solve the QLM scheme (15) to (18) (for more details, refer to the works of Mosta et al [12]). The initial guesses to start the SQLM scheme for the system of equations (15) -(18) are chosen as functions that satisfy the boundary conditions as follows:…”