Computing high-order derivatives in compact integrated-RBF stencils

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.

Automated summary

The study examines the numerical performance of several approximation schemes based on one-dimensional IRBFs for computing high-order derivatives, showing significant improvement in solution accuracy by including nodal values of high-order derivatives. By utilizing extended precision floating point arithmetic, overlapping domain decomposition, and mixed-precision calculations, the efficiency of the approximation schemes is enhanced. The proposed 1D-IRBFs with fixed and variable RBF widths achieve high rates of convergence in solving differential problems and simulating convection-diffusion equations for highly-nonlinear flows.