Improving the accuracy of meshless methods via resolving power optimisation using multiple kernels

TL;DR

Improved resolution characteristics provided by the optimisation procedure presented herein enable accurate simulation of systems of PDEs whose solution contains short spatial scales such as flow fields with homogeneous isotropic turbulence.

Abstract

Meshless methods are commonly used to determine numerical solutions to partial differential equations (PDEs) for problems involving free surfaces and/or complex geometries, approximating spatial derivatives at collocation points via local kernels with a finite size. Despite their common use in turbulent flow simulations, the accuracy of meshless methods has typically been assessed using their convergence characteristics resulting from the polynomial consistency of approximations to operators, with little to no attention paid to the resolving power of the approximation. Here we provide a framework for the optimisation of resolving power by exploiting the non-uniqueness of kernels to provide improvements to numerical approximations of spatial derivatives. We first demonstrate that, unlike in finite-difference approximations, the resolving power of meshless methods is dependent not only on the magnitude of the wavenumber, but also its orientation, before using linear combinations of kernels to maximise resolving power over a range of wavenumbers. The new approach shows improved accuracy in convergence tests and has little impact on stability of time-dependent problems for a range of Eulerian meshless methods. Solutions to a variety of PDE systems are computed, with significant gains in accuracy for no extra computational cost per timestep in Eulerian frameworks. The improved resolution characteristics provided by the optimisation procedure presented herein enable accurate simulation of systems of PDEs whose solution contains short spatial scales such as flow fields with homogeneous isotropic turbulence.

Figures (18)

• Figure 1: Resolving power of the gradient operator for various meshless methods. Left panel $\Re\{k_{eff}\}$, right panel $\Im\{k_{eff}\}$. Orange lines denote $l=2k$, red $l=k$, blue $l=0$, black spectral accuracy. $\cdots$ SPH with Wendland C2 kernel $h/s=1.3$, $\cdot-\cdot$$m=2$ RBF-FDs with Gaussian RBF $\lvert\mathcal{N}_i\rvert=20$, --- $m=4$ LABFM using Hermite polynomials and Wendland C2 kernel. Grey-scale dashed lines denote $2^{\mathrm{nd}}$, $4^{\mathrm{th}}$, and $6^{\mathrm{th}}$ order FDs for comparison (increasing resolving power corresponds to higher order). Left inset shows semi-log plot of $\hat{k}$ against $\lvert \hat{k}-\Re\{\hat{k}_{eff}\}\rvert$, for RBF-FDs. Right inset shows zoomed in graph of $\Im\{\hat{k}_{eff}\}$ for the range $0\leq \hat{k}\leq 1/3$.
• Figure 2: Resolving power graphs of Laplace operator for various meshless methods. Left panel $\Re\{q_{eff}\}$, right panel $\Im\{q_{eff}\}$. Orange lines denote $l=2k$, red $l=k$, blue $l=0$, black spectral accuracy. $\cdots$ SPH with Wendland C2 kernel $h/s=1.3$, $\cdot-\cdot$$m=2$ RBF-FDs with Gaussian RBF $\lvert\mathcal{N}_i\rvert=20$, --- $m=4$ LABFM using Hermite polynomials and Wendland C2 kernel. Grey-scale dashed lines denote $2^{\mathrm{nd}}$, $4^{\mathrm{th}}$, and $6^{\mathrm{th}}$ order FDs for comparison (increasing resolving power corresponds to higher order). Left inset shows semi-log plot of $\hat{q}$ against $\lvert \hat{kq}^2-\Re\{\hat{q}_{eff}^2\}\rvert$, for RBF-FDs. Right inset shows zoomed in graph of $\Im\{\hat{q}_{eff}^2\}$ for the range $0\leq \hat{q}\leq 1/3$.
• Figure 3: Ratio $\mathcal{R}$ of errors arising from SK and MK approach of §\ref{['subsec::Optimisation_of_coefficients']} for SPH. Gradient (left) and Laplacian (right) approximations of \ref{['eq::test_function']}. Red $+$, -- $h/s=2.0$; blue $*$, $--$$h/s=1.6$; black $\triangleleft$, $\cdot-\cdot$$h/s=1.3$. Insets show convergence of $L_2-$ norm of new approximations.
• Figure 4: Ratio $\mathcal{R}$ of errors arising from standard implementation and new approach of §\ref{['subsec::Optimisation_of_coefficients']} for RBF-FDs. Gradient (left) and Laplacian (right) approximations of \ref{['eq::test_function']}. Black $\triangleleft$, $\cdot-\cdot$$m=0$, $\lvert\mathcal{N}_i\rvert=10$; pink $+$, $\cdots$$m=1$, $\lvert\mathcal{N}_i\rvert=15$; blue $*$, $--$$m=2$, $\lvert\mathcal{N}_i\rvert=20$; red $\circ$ , --$m=3$, $\lvert\mathcal{N}_i\rvert=25$. Insets show convergence of $L_2-$ norm of new approximations.
• Figure 5: Ratio $\mathcal{R}$ of errors arising from standard implementation and new approach of §\ref{['subsec::Optimisation_of_coefficients']} for LABFM. Gradient (left) and Laplacian (right) approximations of \ref{['eq::test_function']}. Black $\triangleleft$$m=4$; blue $*$$m=6$; red $\circ$$m=8$. Insets show convergence of $L_2-$ norm of new approximations.