Computing high-order derivatives in compact integrated-RBF stencils

Mai-Duy, N. and Strunin, D. and Karunasena, W. ORCID: (2022) Computing high-order derivatives in compact integrated-RBF stencils. Engineering Analysis with Boundary Elements, 135. pp. 369-381. ISSN 0955-7997

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In Mai-Duy and Strunin (Mai-Duy and Strunin, 2021), it was shown that the inclusion of nodal values of high-order derivatives in compact local integrated-radial-basis-function (IRBF) stencils results in a significant improvement in the solution accuracy. The purpose of this work is to examine in detail the numerical performance of several approximation schemes based on one-dimensional IRBFs for computing high-order derivatives along the grid lines. The extended precision floating point arithmetic is utilised to achieve a high level of accuracy, and the efficiencies of the approximation schemes are improved by employing overlapping domain decomposition and mixed-precision calculations. In solving partial differential equations (PDEs), the proposed 1D-IRBFs are implemented using the RBF widths that are fixed and vary with grid refinement. A simple framework is presented to cover the two RBF width cases, and a numerical analysis is carried out for differential problems with slow and rapid variations in their solutions. In solving the convection–diffusion equations, the proposed 1D-IRBFs are also incorporated into the upwind schemes for effectively simulating highly-nonlinear flows. Numerical results show that high rates of convergence with respect to grid refinement are achieved with both fixed and variable widths.

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Item Type: Article (Commonwealth Reporting Category C)
Refereed: Yes
Item Status: Live Archive
Faculty/School / Institute/Centre: Current – Faculty of Health, Engineering and Sciences - School of Engineering (1 Jan 2022 -)
Faculty/School / Institute/Centre: Current - Institute for Advanced Engineering and Space Sciences (1 Aug 2018 -)
Date Deposited: 23 Feb 2022 04:27
Last Modified: 23 Feb 2022 04:27
Uncontrolled Keywords: High-order derivatives; High-order upwind schemes; Compact approximations; Integrated radial basis functions; Two-dimensional 5-point stencils; RBF widths
Fields of Research (2008): 01 Mathematical Sciences > 0103 Numerical and Computational Mathematics > 010399 Numerical and Computational Mathematics not elsewhere classified
09 Engineering > 0913 Mechanical Engineering > 091307 Numerical Modelling and Mechanical Characterisation
Fields of Research (2020): 40 ENGINEERING > 4017 Mechanical engineering > 401706 Numerical modelling and mechanical characterisation
49 MATHEMATICAL SCIENCES > 4903 Numerical and computational mathematics > 490399 Numerical and computational mathematics not elsewhere classified
Socio-Economic Objectives (2008): E Expanding Knowledge > 97 Expanding Knowledge > 970109 Expanding Knowledge in Engineering
E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences
Socio-Economic Objectives (2020): 28 EXPANDING KNOWLEDGE > 2801 Expanding knowledge > 280110 Expanding knowledge in engineering
28 EXPANDING KNOWLEDGE > 2801 Expanding knowledge > 280118 Expanding knowledge in the mathematical sciences
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