Comparative study on higher order compact RBF-FD formulas with Gaussian and Multiquadric radial functions

TL;DR

This work tackles instability in radial-basis finite-difference methods by deriving analytical weights for Gaussian-based RBF-HFD formulas targeting first and second derivatives up to order $10$ and the 2D-Laplacian up to order $6$. By combining symmetry-aware stencil design with Taylor expansions and symbolic computation, it provides explicit weights and local truncation error expressions, and proves that in the flat limit $\epsilon \to 0$ these weights converge to classical compact FD weights. The authors demonstrate that Gaussian-based RBF-HFD formulas can offer higher accuracy than both multiquadric-based RBF-HFD and traditional compact FD schemes for selected test functions, and they develop a shape-parameter optimization approach to identify $\epsilon^*$ that minimizes LTE. Collectively, the results deliver high-order, stable, analytically tractable RBF-FD schemes with practical guidance for parameter tuning and performance evaluation in PDE discretization.

Abstract

We generate Gaussian radial function based higher order compact RBF-FD formulas for some differential operators. Analytical expressions for weights associated to first and second derivative formulas (up to order 10) and 2D-Laplacian formulas (up to order 6) are derived. Then these weights are used to obtain analytical expression for local truncation errors. The weights are obtained by symbolic computation of a linear system in Mathematica. Often such linear systems are not directly amenable to symbolic computation. We make use of symmetry of formula stencil along with Taylor series expansions for performing the computation. In the flat limit, the formulas converge to their respective order polynomial based compact FD formulas. We validate the formulas with standard test functions and demonstrate improvement in approximation accuracy with respect to corresponding order multiquadric based compact RBF-FD formulas and compact FD schemes. We also compute optimal value of shape parameter for each formula.

Figures (7)

• Figure 1: Shape parameter ($\epsilon$) dependence on absolute LTEs ($|\tau_{0}|$) of RBF-HFD (order 10) formula for first derivative approximation of test functions $u_{1}$ and $u_{2}$ for $h = 0.2,\;0.1,\; 0.05,\; 0.02,\;0.01,\;0.005$ and $0.002$, from top to bottom.
• Figure 2: Comparison of shape parameter ($\epsilon$) dependence on absolute LTEs ($|\tau_{0}|$) of MQ Satya_RBF_HFD and GA based RBF-HFD (order 4, 6, 8 and 10) and respective order compact FD schemes collatznumericallele1992compact for first derivative approximation of test function $u_{1}$ at $x_0=0.4$ and $u_{2}$ at $x_0=0.25$.
• Figure 3: Shape parameter ($\epsilon$) dependence on absolute LTEs ($|\tau_{0}|$) of RBF-HFD (order 10) formula for second derivative approximation of test functions $u_{1}$ and $u_{2}$ and $h = 0.2,\;0.1,\; 0.05,\; 0.02,\;0.01,\;0.005$ and $0.002$, from top to bottom.
• Figure 4: Comparison of shape parameter ($\epsilon$) dependence on absolute LTEs ($|\tau_{0}|$) of MQ Satya_RBF_HFD and GA based RBF-HFD (order 4, 6, 8 and 10) and respective order compact FD schemes collatznumericallele1992compact for second derivative approximations of test function $u_{1}$ at $x_0=0.4$ and $u_{2}$ at $x_0=0.25$.
• Figure 5: Shape parameter ($\epsilon$) dependence on absolute LTEs ($|\tau_{0}|$) of RBF-HFD (order 4) formulas for 2D-Laplacian approximation of test functions (\ref{['Test_function4_Laplacian']}) and (\ref{['Test_function9_Laplacian']}) with step sizes, $h = 0.2,\;0.1,\; 0.05,\; 0.02,\;0.01,\;0.005$ and $0.002$, from top to bottom.