A modified nodal scheme for the time-dependent, incompressible Navier–Stokes equations

Wang, Fei; Rizwan-uddin,

doi:10.1016/s0021-9991(03)00093-7

Cited by 23 publications

(28 citation statements)

References 23 publications

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“…These approximations are commonly used in the NIM [10][11][12] and have been shown to be consistent with the order of the numerical scheme [10]. It is also pointed out that the terms in Equation (11), for which the integration and differential operators do not commute, are kept on the right-hand side of the equation as the pseudo-source terms [10][11][12].…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 70%

“…It is also pointed out that the terms in Equation (11), for which the integration and differential operators do not commute, are kept on the right-hand side of the equation as the pseudo-source terms [10][11][12].…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

“…It should be noted from Equation (10) that f 3 and f 4 are always non-zero resulting in non-zero values of f 30 and f 40 in Equation (11). The coefficients f 10 and f 20 , however, may become zero depending on the geometry of the original element.…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

“…Similar equations can be written for the second ODE in Equation (11). Equation (16) and its counterpart in s direction are written for each element in terms of unknown pseudo-source termsS s0 1 andS t0 2 ; and unknown edge-averaged valuesT…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

“…The solution of these equations yields edge-averaged temperature for all the edges in the domain. Note that NIM yields edge-averaged values instead of the point values [10][11][12].…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

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A nodal integral method for quadrilateral elements

Nezami

Singh

Sobh

et al. 2008

Numerical Methods in Fluids

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SUMMARYNodal integral methods (NIMs) have been developed and successfully used to numerically solve several problems in science and engineering. The fact that accurate solutions can be obtained on relatively coarse mesh sizes, makes NIMs a powerful numerical scheme to solve partial differential equations. However, transverse integration procedure, a step required in the NIMs, limits its applications to brick-like cells, and thus hinders its application to complex geometries. To fully exploit the potential of this powerful approach, abovementioned limitation is relaxed in this work by first using algebraic transformation to map the arbitrarily shaped quadrilaterals, used to mesh the arbitrarily shaped domain, into rectangles. The governing equations are also transformed. The transformed equations are then solved using the standard NIM. The scheme is developed for the Poisson equation as well as for the time-dependent convectiondiffusion equation. The approach developed here is validated by solving several benchmark problems. Results show that the NIM coupled with an algebraic transformation retains the coarse mesh properties of the original NIM.

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Section: Formulation For 2d Poisson Equationmentioning

confidence: 70%

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

“…The solution of these equations yields edge-averaged temperature for all the edges in the domain. Note that NIM yields edge-averaged values instead of the point values [10][11][12].…”

Section: Formulation For 2d Poisson Equationmentioning

confidence: 99%

See 3 more Smart Citations

A nodal integral method for quadrilateral elements

Nezami

Singh

Sobh

et al. 2008

Numerical Methods in Fluids

View full text Add to dashboard Cite

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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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