A paired spectral-finite difference approach for solving boundary layer flow problems

Abstract

This study presents an innovative numerical technique for obtaining solutions to systems of partial differential equations. Two previously introduced numerical methods, namely the spectral quasilinearization method (SQLM) (Motsa, in J Appl Math 2013:Article ID 423628, 2013) and the spectral local linearization method (SLLM) (Motsa 2013) have been shown to be efficient methods for solving systems of PDEs but both contain slight limitations. Our proposed method hereinafter referred to as the paired spectral quasilinearization method (PSQLM), seeks to simplify a large system of PDEs by decoupling it into pairs of sub-systems and applying quasilinearization on the pairs in order to linearize the pairs. Crank–Nicolson scheme and spectral collocation method are, respectively, applied to discretize the system of decoupled linearized pairs of PDEs. The PSQLM is described for a general system of PDEs and a numerical experiment is conducted on a system of four differential equations that contain three possible combinations. We compare the different combinations to ascertain the effect of performing different pairings on the accuracy and convergence speed of solutions. We also compare the PSQLM with the SQLM (which solves the system as coupled) and the SLLM (that decouples a system) to test the efficacy of the PSQLM. The observations made from the comparison proves the PSQLM converges faster than the SLLM with very few iterations. It is also shown to use significantly lesser computational time than the SQLM to generate convergent solutions. The accuracy of the PSQLM is also observed to better both the SQLM and SLLM.