Optimization, resolution and application of composite compact finite difference templates

Jordan, Stephen A.

doi:10.1016/j.apnum.2010.08.008

Cited by 3 publications

(8 citation statements)

References 23 publications

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“…Another important quality, particularly for aeroacoustic simulations is that the final error tends to zero after the wave has left the domain. In this regard the result provided by the current schemes again outperforms that of previous studies resulting in a final error reduction of 97.8% compared to Kim [1], 98.9% compared to Jordan [13] and 96.9% compared to Liu et al [11]. Figure 13 shows the maximum ℓ 2 -norm errors produced by each scheme at various grid levels.…”

Section: One-dimensional Scalar Wavementioning

confidence: 46%

“…The present schemes exclusively exhibit no overshoot as the wave leaves the domain. This corresponds to peak error reductions of 38.4%, 66.1% and 84.1% compared to that produced by the coefficients of Kim [1], Jordan [13] and Liu et al [11], respectively. Another important quality, particularly for aeroacoustic simulations is that the final error tends to zero after the wave has left the domain.…”

Section: One-dimensional Scalar Wavementioning

confidence: 99%

“…The most robust performance is attained by the new schemes, which maintain similar error levels with and without filtering. In fact, they are the only schemes for which the peak error is slightly increased by filtering, suggesting that they are success- [1] and Jordan [13] extra caution should be exercised while selecting the filter cut-off wavenumber. The maximum ℓ 2 -norm errors produced by the new schemes with and without filtering is shown in Figure 15.…”

Section: One-dimensional Scalar Wavementioning

confidence: 99%

“…In past procedures the numerical stability is often treated as an afterthought, and is only considered after the wavenumber optimisation of the finite difference coefficients is completed ( [1,[11][12][13]). …”

Section: Linear Stability Constraintmentioning

confidence: 99%

“…Jordan applied this analysis to tridiagonal systems, employing a least squares optimisation strategy to minimise the total resolution error across the whole template. In a later paper by Jordan [13] the same technique was applied to pentadiagonal systems producing a set a of boundary closure schemes to be used alongside the interior scheme of Kim [1]. Although the modified wavenumber curves produced by this technique are dependent on the number of grid points used in the analysis, they appear to be much more representative of the performance we achieve once schemes are applied to actual simulations.…”

Section: Introductionmentioning

confidence: 99%

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Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

Turner

Haeri²,

Kim

2016

Computers & Fluids

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This paper presents efforts to improve the boundary efficiency and accuracy of a compact finite difference scheme, based on its composite template.Unlike precursory attempts the current methodology is unique in its quantification of dispersion and dissipation errors, which are only evaluated after the matrix system of equations has been rearranged for the derivative. This results in a more accurate prediction of the boundary performance, since the analysis is directly based on how the derivative is represented in simulations.A genetic algorithm acts as a comprehensive method for the optimisation of the boundary coefficients, incorporating an eigenvalue constraint for the linear stability of the matrix system of equations. The performance of the optimised composite template is tested on one-dimensional linear wave convection and two-dimensional inviscid vortex convection problems, with uniform and curvilinear grids. In all cases, it yields substantial accuracy and * Corresponding author, Email address: J.M.Turner@soton.ac.uk * * Current address: Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glassgow, G1 1XQ, UK Preprint submitted to Computers & Fluids August 16, 2016 efficiency improvements while maintaining stable solutions and fourth-order accuracy.

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Section: One-dimensional Scalar Wavementioning

confidence: 46%

Section: One-dimensional Scalar Wavementioning

confidence: 99%

Section: One-dimensional Scalar Wavementioning

confidence: 99%

Section: Linear Stability Constraintmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

Turner

Haeri²,

Kim

2016

Computers & Fluids

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Towards a genuinely stable boundary closure for pentadiagonal compact finite difference schemes

Wu,

Kim

2024

Journal of Computational Physics

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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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Optimization, resolution and application of composite compact finite difference templates

Cited by 3 publications

References 23 publications

Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template

Towards a genuinely stable boundary closure for pentadiagonal compact finite difference schemes

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

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