Summary of Modern Nodal Integral Methods in Fluid Flow and Heat Transfer

Wilson, Gary L.; Rydin, R.A.

doi:10.1007/978-3-642-84238-2_71

Cited by 3 publications

(1 citation statement)

References 8 publications

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“…10,12,13 A few years later, Esser and Witt 14 gave a NIM scheme for the steady-state two-dimensional Navier-Stokes equation and showed that the scheme has inherent upwinding. NIM was then developed and applied to several transient problems, such as heat conduction, 15 convection-diffusion, 16 and Navier-Stokes problems, 14,17 to achieve the same inherency in time. An earlier NIM application for time-dependent problems approximated the time derivative by a simple Euler scheme, due to which NIM could not be utilized to its full potential.…”

Section: Introductionmentioning

confidence: 99%

Physics‐based preconditioning of Jacobian‐free Newton–Krylov solver for Navier–Stokes equations using nodal integral method

Ahmed,

Singh,

Kumar

2023

Numerical Methods in Fluids

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The nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M2NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M2NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobian‐free Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M2NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the time‐splitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.

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