arXiv subjects

Andrej Kolar-Požun

Publications and source records attributed to Andrej Kolar-Požun.

8 recordsLinked to original sources

arXiv2026

Adaptive hyperviscosity stabilisation for the RBF-FD method in solving advection-dominated transport equations

This paper presents an adaptive hyperviscosity stabilisation procedure for the Radial Basis Function-generated Finite Difference (RBF-FD) method, aimed at solving linear and non-linear advection-dominated transport equations on domains without a boundary. The approach employs a PDE-independent algorithm that adaptively determines the hyperviscosity constant based on the spectral radius of the RBF-FD evolution matrix. The proposed procedure supports general node layouts and is not tailored for specific equations, avoiding the limitations of empirical tuning and von Neumann-based estimates. To reduce computational cost, it is shown that lower monomial augmentation in the approximation of the hyperviscosity operator can still ensure consistent stabilisation, enabling the use of smaller stencils and improving overall efficiency. A hybrid strategy employing different spline orders for the advection and hyperviscosity operators is also implemented to enhance stability. The method is evaluated on pure linear advection and non-linear Burgers' equation, demonstrating stable performance with limited numerical dissipation. The two main contributions are: (1) a general hyperviscosity RBF-FD solution procedure demonstrated on both linear and non-linear advection-dominated problems, and (2) an in-depth analysis of the behaviour of hyperviscosity within the RBF-FD framework, addressing the interplay between key free parameters and their influence on numerical results.

math.NA↗

arXiv2026

A representation and comparison of three cubic macro-elements

The paper is concerned with three types of cubic splines over a triangulation that are characterized by three degrees of freedom associated with each vertex of the triangulation. The splines differ in computational complexity, polynomial reproduction properties, and smoothness. With the aim to make them a versatile tool for numerical analysis, a unified representation in terms of locally supported basis functions is established. The construction of these functions is based on geometric concepts and is expressed in the Bernstein--Bézier form. They are readily applicable in a range of standard approximation methods, which is demonstrated by a number of numerical experiments.

math.NA↗

arXiv2026

An RBF-based method for computational electromagnetics with reduced numerical dispersion

The finite difference time domain method is one of the simplest and most popular methods in computational electromagnetics. This work considers two possible ways of generalising it to a meshless setting by employing local radial basis function interpolation. The resulting methods remain fully explicit and are convergent if properly chosen hyperviscosity terms are added to the update equations. We demonstrate that increasing the stencil size of the approximation has a desirable effect on numerical dispersion. Furthermore, our proposed methods can exhibit a decreased dispersion anisotropy compared to the finite difference time domain method.

physics.comp-ph↗

arXiv2026

Oscillatory behaviour of the RBF-FD approximation accuracy under increasing stencil size

When solving partial differential equations on scattered nodes using the Radial Basis Function generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the signed approximation error. Based on this observation we are then able to introduce a numerical quantity that indicates whether a given stencil size is locally optimal.

math.NA↗

arXiv2026

Some observations regarding the RBF-FD approximation accuracy dependence on stencil size

When solving partial differential equations on scattered nodes using the Radial Basis Function-generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the signed approximation error. Based on this observation we are able to introduce a numerical quantity that could indicate whether a given stencil size is locally optimal. This work is an extension of our ICCS 2023 conference paper.

math.NA↗

arXiv2026

A superconvergence result in the RBF-FD method

Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is well known that the truncation error of the differential operator approximation is determined by the degree of monomial augmentation. Naively, one might think that the solution error will have the same order of convergence. We present a superconvergence result that shows otherwise - for some augmentation degrees, order of convergence is higher than expected.

math.NA↗

arXiv2026

A Numerical Study of Combining RBF Interpolation and Finite Differences to Approximate Differential Operators

This paper focuses on RBF-based meshless methods for approximating differential operators, one of the most popular being RBF-FD. Recently, a hybrid approach was introduced that combines RBF interpolation and traditional finite difference stencils. We compare the accuracy of this method and RBF-FD on a two-dimensional Poisson problem for standard five-point and nine-point stencils and different method parameters.

math.NA↗

arXiv2026

Computational Electromagnetics with the RBF-FD Method

One of the most popular methods employed in computational electromagnetics is the Finite Difference Time Domain (FDTD) method. We generalise it to a meshless setting using the Radial Basis Function generated Finite Difference (RBF-FD) method and investigate its properties on a simple test problem.

physics.comp-ph↗