On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

Flyer, Natasha; Fornberg, Bengt; Bayona, Víctor; Barnett, G. A.

doi:10.1016/j.jcp.2016.05.026

Cited by 241 publications

(276 citation statements)

References 32 publications

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“…Depending on the application, it may be interesting to consider interpolants in which the degree of the polynomial part is arbitrarily high. It has been shown in [11] that higher accuracy can be achieved for problems on circular domains by introducing supplementary polynomials. It is also observed that small shape parameters can then be used even without employing stable algorithms.…”

Section: Infinitely Smooth Rbfsmentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley-Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.

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Section: Infinitely Smooth Rbfsmentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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“…This results in a less oscillatory interpolant and thus more accurate derivative approximations. Further augmentation with more polynomials is currently being studied in Flyer et al (2015a) and Bayona et al (2015). As a result, the system of equations that determines the RBF-FD differentiation weight w i to approximate Lu is…”

Section: Rbf-fd Methodsmentioning

confidence: 99%

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

et al. 2015

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Abstract. A numerical model based on radial basis functiongenerated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linear elliptic partial differential equation (PDE) in a spherically shaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To our knowledge, this is (1) the first numerical model of the GEC to combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this, (2) the first RBF-FD method to use irregular 3-D stencils for discretization to handle the topography. It benefits from the mesh-free nature of RBF-FD, which is especially suitable for modeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes no limiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the right hand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.

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“…We can rewrite (14) so that it explicitly depends only on the vector of samples f S1 using (9) as follows: (15) (…”

Section: Computation Of the Weights From Iterated Differentiation Thmentioning

confidence: 99%

“…There are workarounds for both of these problems. In the stationary interpolation case, it is possible to recover high-order convergence by adding suitable polynomial terms to the interpolant [14,3]. On a surface, however, the polynomials may themselves introduce ill-conditioning, especially if it is an algebraic surface as polynomial unisolvency becomes an issue.…”

Section: Parameter Studiesmentioning

confidence: 99%

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

Lehto¹,

Shankar²,

Wright³

2017

SIAM J. Sci. Comput.

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Abstract. We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in R d . The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds.

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On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

Cited by 241 publications

References 32 publications

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

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