Minimal numerical differentiation formulas

Davydov, Oleg; Schaback, Robert

doi:10.1007/s00211-018-0973-3

Cited by 31 publications

(42 citation statements)

References 25 publications

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“…The automatic selection of δ based on stability properties of augmented local RBF interpolants is a subject of ongoing research. It is likely that the optimal choice of δ depends on the local Lebesgue functions corresponding to the differential operator L; see Section 3 and [24,29] for discussions on these local Lebesgue functions.…”

Section: Parameter Selectionmentioning

confidence: 99%

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Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Shankar

Fogelson

2018

Journal of Computational Physics

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We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advectiondiffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit-explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

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Section: Parameter Selectionmentioning

confidence: 99%

“…end if 17: end for where y is some evaluation point, h(y) is the largest distance between the point y and every point x in the stencils, P (y) is some growth function [29], and α is a multiindex. The term max |α|= D α f W 1,∞ (Ω1) is simply the Sobolev ∞-norm of f on the stencil P 1 with convex hull Ω 1 .…”

mentioning

confidence: 99%

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Shankar

Fogelson

2018

Journal of Computational Physics

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“…In general, one can use the M given nodes for getting exactness on polynomials of maximal order, and then there can be additional degrees of freedom because the Q × M linear system (6) may be nonuniquely solvable. The paper [13] deals with various techniques to use the additional degrees of freedom, e.g. for minimizing the ℓ 1 norm of the weights.…”

Section: Optimal Convergence On Polynomialsmentioning

confidence: 99%

“…By solving the system (6), such stencils are easy to calculate, but if the system is underdetermined, one should make good use of the additional degrees of freedom. This topic is treated in [13] by applying optimization techniques, while the next sections will focus on unique stencils obtained by polyharmonic kernels. Because the latter come close to the kernels reproducing Sobolev spaces, they should provide good approximations to the non-scalable optimal approximations in Sobolev spaces.…”

Section: Optimal Convergence On Polynomialsmentioning

confidence: 99%

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Optimal stencils in Sobolev spaces

Davydov

Schaback

2017

IMA Journal of Numerical Analysis

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This paper proves that the approximation of pointwise derivatives of order s of functions in Sobolev space W m 2 (R d ) by linear combinations of function values cannot have a convergence rate better than m − s − d/2, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by scalable approximations that are exact on polynomials of order at least ⌊m − d/2⌋ + 1, proving that the rates are optimal for given m, s, and d. And, for a fixed node set X ⊂ R d , the convergence rate in any Sobolev space W m 2 (Ω) cannot be better than q − s where q is the maximal possible order of polynomial exactness of approximations based on X , no matter how large m is. In particular, scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences (RBF-FD). Numerical examples are included for illustration. 1 Univ. Gießen, Oleg.Davydov@math.uni-giessen.de https://www.staff.uni-giessen.de/odavydov/ 2 Univ. Göttingen, schaback@math.uni-goettingen.de

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Approximation with Conditionally Positive Definite Kernels on Deficient Sets

Davydov

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Minimal numerical differentiation formulas

Cited by 31 publications

References 25 publications

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Optimal stencils in Sobolev spaces

Approximation with Conditionally Positive Definite Kernels on Deficient Sets

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