Picard and Newton-Based Preconditioned Nodal Integral Method for the Solution of Fluid Flow Equations

The nuclear industry has relied on the Nodal Integral Methods (NIMs) for many years as a workhorse. Despite their significant development, the fluid flow community has only a limited level of acceptance for these methods. One considerable drawback with these approaches for solving fluid flow problems is the absence of reliable and effective nonlinear solvers for the algebraic equations generated during the discretization procedure. Recently, some efforts have been made for fluid flow problems to extend the application of NIM for higher nonlinearity by employing the Jacobian-free Newton–Krylov (JFNK) technique, which is an efficient Newton method implementation. In NIM, there is a slight difference between Picard’s and Newton’s implementations; therefore, knowing the required steps for implementing these approaches with NIM is also essential. Preconditioning is usually necessary for both methods for a computationally efficient solution. In the present work, it is realized that the preconditioner developed earlier for Burgers’ equation is more effective in the case of Picard’s method than in Newton’s method. Therefore, the present study compares the implementation process of the Preconditioned Picard and Preconditioned Newton-based approaches within the framework of the NIM. Additionally, spectral analysis and comparisons of nonlinear iterations for both systems are provided to illustrate the convergence of these methods.