A nodal coarse-mesh method for the efficient numerical solution of laminar flow problems

Horak, W.C.; Dorning, J.J.

doi:10.1016/0021-9991(85)90120-2

Cited by 17 publications

(8 citation statements)

References 14 publications

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“…Coarse-mesh nodal methods, including nodal integral methods (NIMs), have been developed over the last two decades to efficiently solve sets of linear and nonlinear PDEs [5][6][7][8][9][10][11][12]4]. Among the distinguishing features of some of the coarse mesh methods is the transverse integration process that-after the problem domain has been divided into nodes-reduces each of the PDE to an ODE by integrating over the node dimensions of all-but-one independent variables [6][7][8]10].…”

Section: Background and Motivationmentioning

confidence: 99%

“…Among the distinguishing features of some of the coarse mesh methods is the transverse integration process that-after the problem domain has been divided into nodes-reduces each of the PDE to an ODE by integrating over the node dimensions of all-but-one independent variables [6][7][8]10]. Carrying out the procedure repeatedly, one obtains a set of m ODEs for each PDE, where m is the number of the independent variables.…”

Section: Background and Motivationmentioning

confidence: 99%

“…Equations (2) and (6)(7)(8) in the main text of this article do not change for the conventional NIM. Equation (3) of the main text is written differently in the conventional method-absorbing the nonlinear term in the pseudo-source termS t 2 (x):…”

Section: Appendix A: Application Of the Conventional Nodal Integral Mmentioning

confidence: 99%

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An improved coarse-mesh nodal integral method for partial differential equations

Rizwan-uddin

1997

Numer. Methods Partial Differential Eq.

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An improved variation of the nodal integral method to solve partial differential equations has been developed and implemented. Rather than treating all of the nonlinear terms as the so-called pseudo-source terms (to be approximated), in this modified version of the nodal integral method, by approximating part of the nonlinear terms in terms of the discrete variable(s) that ultimately result at the end of the formulation process, some or all of the nonlinear terms are kept on the left-hand side in the transverse-integrated equations, which are to be solved analytically. Application of the method to solve the Burgers equation leads to exponential variation within the nodes and shows that the resulting scheme has inherent upwinding. Reconstruction of node interior solution -as a function of one independent variable, and averaged in all others -makes it possible to obtain rather accurate solutions even on a fine scale. Results of the numerical analysis and comparison with results of other methods reported in the literature show that the new method is comparable and sometimes better in accuracy than the currently used schemes. Extension to multidimensional problems is straightforward.

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Section: Background and Motivationmentioning

confidence: 99%

Section: Background and Motivationmentioning

confidence: 99%

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An improved coarse-mesh nodal integral method for partial differential equations

Rizwan-uddin

1997

Numer. Methods Partial Differential Eq.

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“…NIM schemes have been developed and applied to solve fluid flow and heat transfer problems. In earlier implementations of this method, schemes were developed by clubbing the nonlinear convection terms of Navier-Stokes equations into inhomogeneous terms of ODEs generated from N-S equations [13][14][15][16]. This leads to an inefficient generation of shape functions for velocity components that captures only the diffusion process and not convection.…”

Section: Introductionmentioning

confidence: 98%

“…This leads to an inefficient generation of shape functions for velocity components that captures only the diffusion process and not convection. Moreover, use of the continuity equation to develop cell analytical solution leads to asymmetry in local solutions or shape functions of primitive variables (velocity components) in a cell [13][14][15][16]. The shortfalls of these schemes were addressed by the modified nodal integral method (MNIM) proposed by Rizwan-uddin [17].…”

Section: Introductionmentioning

confidence: 99%

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

Kumar

Singh

Doshi

2012

Numerical Heat Transfer, Part B: Fundamentals

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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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A nodal coarse-mesh method for the efficient numerical solution of laminar flow problems

Cited by 17 publications

References 14 publications

An improved coarse-mesh nodal integral method for partial differential equations

An improved coarse-mesh nodal integral method for partial differential equations

Pressure Correction–Based Iterative Scheme for Navier-Stokes Equations using Nodal Integral Method

A nodal integral method for quadrilateral elements

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