A Least Squares Radial Basis Function Finite Difference Method with Improved Stability Properties

Tominec, Igor; Larsson, Elisabeth; Heryudono, Alfa

doi:10.1137/20m1320079

Cited by 35 publications

(33 citation statements)

References 27 publications

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“…When M > N , the discretization is oversampled: E h and D h are rectangular. In order to simplify the theoretical analysis at a later stage in this paper, we introduce a norm scaling of (2.3) for a 2D problem [25], and define Ēh = h y E h and Dh = h y D h , to obtain:…”

Section: Point Set Xmentioning

confidence: 99%

“…) is a discrete inner product. The 2 inner product is an approximation of the L 2 inner product (u, v) L2 = Ω uv dΩ [25]. Interestingly, if we had initially projected (1.1) using the L 2 inner product, instead of employing oversampling and a projection that led to the 2 inner product, the final discretization would have yielded a pure Galerkin scheme.…”

Section: Point Set Xmentioning

confidence: 99%

“…Substituting (6.2) into the left hand side of the estimate above, and then using that [25], we have that: (6.21)…”

Section: 4mentioning

confidence: 99%

“…A related work is [25] where we introduced an oversampled version of the RBF-FD method for an elliptic model problem, and made a theoretical investigation of its stability properties, where we showed that the RBF-FD method generates a discontinuous trial space and that oversampling of a stationary problem approximates a variational formulation using a least-squares projection. The oversampled RBF-FD method was then also extended to an unfitted setting in [24], which was later used for the simulations of a thoracic diaphragm in [27].…”

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confidence: 99%

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Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Tominec¹,

Nazarov²,

Larsson³

2021

Preprint

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We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete 2 -norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be 2 -stable in time under a sufficiently large oversampling of the discretized system of equations. On the other hand, the RBF-FD method is not 2 -stable by construction, no matter how large the oversampling is. We show that this is due to the jumps (discontinuities) in the RBF-FD cardinal basis functions. We also provide a stabilization of the RBF-FD method that penalizes the spurious jumps. Numerical experiments show an agreement with our theoretical observations.

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Section: Point Set Xmentioning

confidence: 99%

Section: Point Set Xmentioning

confidence: 99%

“…Substituting (6.2) into the left hand side of the estimate above, and then using that [25], we have that: (6.21)…”

Section: 4mentioning

confidence: 99%

mentioning

confidence: 99%

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Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Tominec¹,

Nazarov²,

Larsson³

2021

Preprint

Self Cite

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show abstract

“…In numerical studies, ε = 0.1 was found to have a good balance between these factors. Applying a least-squares method to RBF-FD has been used on the local stencil level to increase numerical stability, especially for Neumann boundary conditions [23]. Here, a least squared approach is used in terms of applying an LSQR iterative solver to invert the tall (i.e.…”

Section: Solving the Discretized Systemmentioning

confidence: 99%

Solving 3D Magnetohydrostatics with RBF-FD: Applications to the Solar Corona

Mathews,

Flyer,

Gibson

2021

Preprint

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We present a novel magnetohydrostatic numerical model that solves directly for the forcebalanced magnetic field in the solar corona. This model is constructed with Radial Basis Function Finite Differences (RBF-FD), specifically 3D polyharmonic splines plus polynomials, as the core discretization. This set of PDEs is particularly difficult to solve since in the limit of the forcing going to zero it becomes ill-posed with a multitude of solutions. For the forcing equal to zero there are no numerically tractable solutions. For finite forcing, the ability to converge onto a physically viable solution is delicate as will be demonstrated. The static force-balance equations are of a hyperbolic nature, in that information of the magnetic field travels along characteristic surfaces, yet they require an elliptic type solver approach for a sparse overdetermined ill-conditioned system. As an example, we reconstruct a highly nonlinear analytic model designed to represent long-lived magnetic structures observed in the solar corona.

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Oscillatory Behaviour of the RBF-FD Approximation Accuracy Under Increasing Stencil Size

Andrej

Jančič

Rot

et al. 2023

Lecture Notes in Computer Science

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A Least Squares Radial Basis Function Finite Difference Method with Improved Stability Properties

Cited by 35 publications

References 27 publications

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Stability estimates for radial basis function methods applied to time-dependent hyperbolic PDEs

Solving 3D Magnetohydrostatics with RBF-FD: Applications to the Solar Corona

Oscillatory Behaviour of the RBF-FD Approximation Accuracy Under Increasing Stencil Size

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