Predictor-Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers' equation

摘要

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as (MNIM)-N-2), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and (MNIM)-N-2 schemes, respectively, it was observed that although the modified scheme with delayed coefficients ((MNIM)-N-2) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor-corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized (MNIM)-N-2 scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of (MNIM)-N-2 with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection-diffusion problem, represented by the Burgers' equation. The computational results for both one-dimensional and two-dimensional Burgers' equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods. (C) 2019 Elsevier Ltd. All rights reserved.