Applied Numerical Mathematics 161 (2021) 13–26 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum A third-order accurate in time method for boundary layer flow problems Syed Ahmed Pasha a,∗ , Yasir Nawaz b , Muhammad Shoaib Arif b a Department of Electrical & Computer Engineering, Air University, Service Road E-9/E-8, Islamabad 44000, Pakistan b Department of Mathematics, Air University, Service Road E-9/E-8, Islamabad 44000, Pakistan a r t i c l e i n f o a b s t r a c t Article history: Received 14 August 2020 Received in revised form 8 October 2020 Accepted 26 October 2020 Available online 29 October 2020 Keywords: Parabolic partial differential equations Numerical approximation Von Neumann stability Consistency analysis Stokes problem The boundary layer flow problem arises in numerous industrial applications. As a result, it has received considerable attention over the last five decades which has involved developing numerical procedures to approximate the solution of time-dependent parabolic and first-order hyperbolic partial differential equations (PDEs). In this paper, we develop a method that guarantees third-order temporal accuracy. Stability conditions are derived using Von Neumann stability analysis that guarantee convergence of the proposed algorithm. In addition, we present a consistency analysis. The performance of the proposed algorithm is demonstrated for linear and nonlinear parabolic PDEs that extend the models for the Stokes first and second problems by incorporating the effect of heat transfer using viscous dissipation and thermal radiation. A comparison of performance in terms of convergence rate and estimation accuracy is shown.  2020 IMACS. Published by Elsevier B.V. All rights reserved. 1. Introduction The boundary layer flow problem has attracted considerable attention over the last five decades. The problem manifests in a number of industrial applications including continuous casting, polymer extrusion, glass fibre and paper production, stretching of plastic films and food manufacturing. Stokes first problem considers the flow of an unsteady viscous fluid on a flat stretched plate. Stokes second problem is concerned with the flow on an oscillating plate. We begin with a review of selected works on the solution of Stokes problems. In [5], the flow of Oldroyd-B fluid has been studied in the setting of Stokes first problem to assess the tension and velocity field. The authors have shown that the solutions for Maxwell fluid, Navier-Stokes and second-degree fluid are limiting cases of their solution. A relationship between three unsteady viscous flows which included the unsteady boundary layer flow and Stokes first problem has been studied in [10]. An asymptotic analysis has been presented to study the short- time and long-time behaviour of the problems. Stokes first problem for nanofluids has been studied in [19]. The authors have attempted to study the effects of Brownian motion and thermophoresis on the velocity, temperature and nanoparticle volume fractions. The flow of a thermoelectric fluid over a suddenly moving heated surface has been studied in [4] and exact solutions in the Laplace domain have been developed. Stokes first problem with heat transfer has been studied in [13]. In [6], Hyat et al. have studied the flow of a Johnson-Segalman fluid on an oscillating plate and furthermore shown that the results for a viscous fluid and Maxwell fluid are obtained as special cases of their proposed solution. Using the Laplace * Corresponding author. E-mail address: s.pasha@mail.au.edu.pk (S.A. Pasha). https://doi.org/10.1016/j.apnum.2020.10.023 0168-9274/ 2020 IMACS. Published by Elsevier B.V. All rights reserved.