A comparative analysis of numerical methods applied to nonsimilar boundary layer-derived infinite series equations

Abstract

Circa 1958, Merk propounded a boundary layer procedure valid for both similarity and nonsimilarity problems. It was notably the first asymptotic expansion to account for boundary layer nonsimilarity. Due to an unfortunate error in the procedure, the method was later ameliorated by Chao and Fagbenle and is today commonly referred to as the Merk-Chao-Fagbenle (MCF) method. The objective in this work is an investigation to compare two numerical methods—the single-step multistage method known as the fourth-order Runge–Kutta method with the Newton–Raphson shooting iteration as the root-finding algorithm (RK + Newton), and the finite-element method (FEM). In so doing, the characteristic nonsimilar perturbation series boundary layer problem of Merk, Chao, and Fagbenle is employed as a model. The novelty is to assess critical numerical performance indices of both numerical techniques, which constitutes an undertaking that has yet to be elucidated, as far as the authors are aware. Thus, this work departs from the norm and advances beyond previous efforts in literature by emphasizing the numerical performances of two numerical methods rather than the sundry boundary layer solutions, which in any case have been presented in previous works. It is found that the numerical results obtained using both methods correlate very well with highly accurate benchmarked results. The role of each method to evaluate the velocity functions (fs) and temperature functions (θs) is visually depicted and described numerically. The computation and central processing unit (CPU) times for the evaluation of f0,f1,f2,f3, and θ0,θ1,θ2,θ3 according to both the FEM and the RK + Newton methods for element sizes of 10-3 and 10-4 reveal that the computation time of RK + Newton is significantly less than that of the FEM for both values of the element size. On the other hand, the CPU time of RK + Newton is less than that of the FEM for f0, and θ0 only. However, overall, FEM is much more accurate.