Development of an Upwinding Scheme through the Minimization of Modified Wavenumber Error for the Incompressible Navier-Stokes Equations

Sheu, Tony W. H.; Kao, Neo Shih-Chao; Chiu, Pao‐Hsiung; Lin, Chang–Shou

doi:10.1080/10407790.2011.601147

Cited by 5 publications

(2 citation statements)

References 20 publications

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“…Sheu et al [2] applied a three-step fractional step splitting solution algorithm to timedependent Navier-Stokes equations for incompressible fluid for high Reynolds number as well for high Rayleigh number problems. Sheu et al [3] also proposed a convection scheme to get better dispersive accuracy by minimizing the dispersive error by introducing Lagrangian multipliers. Hwang [4][5][6][7] developed a descretization scheme for incompressible Navier-Stokes equations based on an equation-solving solution gradient for an unstructured grid system.…”

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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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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“…However, to give a sense of the variety of DRB methods, we do mention that examples of existing optimized DRB schemes include upwind-type schemes [24,31,41,42], multi-scale and multi-gridsize schemes [4,35], DRB schemes with non-uniform girds for general geometry [9,31], composite boundary schemes [15,16,18,19], optimized prefactored compact schemes [2], and optimized DRB schemes that couple space-time discretization [27,29]. In addition to the development of optimized DRB schemes, there are also comparison studies of the performance of optimized DRB schemes [12,25,26,43], applications of DRB schemes in fluid flow and non-linear waves [10,11,29,30], and error dynamics of optimized DRB schemes [28]. 1.3.…”

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confidence: 99%

Analysis and development of compact finite difference schemes with optimized numerical dispersion relations

Kuo¹,

Lee²,

Lyng³

2014

Preprint

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Finite difference approximation, in addition to Taylor truncation errors, introduces numerical dispersion-and-dissipation errors into numerical solutions of partial differential equations. We analyze a class of finite difference schemes which are designed to minimize these errors (at the expense of formal order of accuracy), and we analyze the interplay between the Taylor truncation errors and the dispersion-and-dissipation errors during mesh refinement. In particular, we study the numerical dispersion relation of the fully discretized non-dispersive transport equation in one and two space dimensions. We derive the numerical phase error and the L 2 -norm error of the solution in terms of the dispersion-and-dissipation error. Based on our analysis, we investigate the error dynamics among various optimized compact schemes and the unoptimized higher-order generalized Padé compact schemes, taking into account four important factors, namely, (i) error tolerance, (ii) computer memory capacity, (iii) resolvable wavenumber, and (iv) CPU/GPU time. The dynamics shed light on the principles of designing suitable optimized compact schemes for a given problem. Using these principles as guidelines, we then propose an optimized scheme that prescribes the numerical dispersion relation before finding the corresponding discretization. This approach produces smaller numerical dispersion-and-dissipation errors for linear and nonlinear problems, compared with the unoptimized higher-order compact schemes and other optimized schemes developed in the literature. Finally, we discuss the difficulty of developing an optimized composite boundary scheme for problems with non-trivial boundary conditions. We propose a composite scheme that introduces a buffer zone to connect an optimized interior scheme and an unoptimized boundary scheme. Our numerical experiments show that this strategy produces small L 2 -norm error when a wave packet passes through the non-periodic boundary.

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Literature Survey of Numerical Heat Transfer (2010–2011)

Shih

Zheng

Arie

et al. 2013

Numerical Heat Transfer, Part A: Applications

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Development of an Upwinding Scheme through the Minimization of Modified Wavenumber Error for the Incompressible Navier-Stokes Equations

Cited by 5 publications

References 20 publications

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

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

Analysis and development of compact finite difference schemes with optimized numerical dispersion relations

Literature Survey of Numerical Heat Transfer (2010–2011)

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