
Spherical basis function approximation with particular trend functions
Segeth, Karel
The paper is concerned with the measurement of scalar physical quantities at nodes on the $(d1)$dimensional unit sphere surface in the hbox{$d$dimensional} Euclidean space and the spherical RBF interpolation of the data obtained. In particular, we consider $d=3$. We employ an inverse multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 defined in Cartesian coordinates. We prove the existence of the interpolation formula of the type considered. The formula can be useful in the interpretation of many physical measurements. We show an example concerned with the measurement of anisotropy of magnetic susceptibility having extensive applications in geosciences and present numerical difficulties connected with the high condition number of the matrix of the system defining the interpolation.


Cosmology on Small Scales 2022: Dark Energy and the Local Hubble Expansion Problem
Křížek, Michal ; Dumin, Y. V.
One hundred years ago, Russian mathematician and physicist Alexander A. Friedmann applied the system of Einstein equations to the threedimensional sphere with a time varying radius. In this way, he obtained a nonlinear ordinary differential equation which is called the Friedmann equation after him and serves now as a cornerstone of the standard cosmological model. Unfortunately, it is well known that this model exhibits a number of paradoxes. Thus, the main goal of the CSS 2022 Conference Proceedings is to discuss whether and how the Friedmann equation can be applied at the various spatial scales, from our local cosmic neighborhood up to the whole Universe, and if the existence of dark matter and dark energy are merely artifacts of the excessive extrapolations.


100 years of the Friedmann equation
Křížek, Michal
In 1922, Alexander Friedmann applied Einstein’s equations to a threedimensional sphere to describe the evolution of our universe. In this way he obtained a nonlinear ordinary differential equation (called after him) for the expansion function representing the radius of that sphere. At present, the standard cosmological ΛCDM model of the universe is based just on the Friedmann equation. It needs a significant amount of dark matter, about six times that of the usual baryonic matter, besides an even larger amount of dark energy to be consistent with the real universe. But to date, both dark matter and dark energy have remained without concrete evidence based on direct physical measurements. We present several arguments showing that such a claimed amount of dark matter and dark energy can only be the result of vast overestimation, incorrect extrapolations, and that it does not correspond to the real universe. The spatial part of our universe seems to be locally flat and thus it can be locally modeled by the Euclidean space. However, Friedmann did not consider the flat space with zero curvature. Therefore, in the second part of this paper we will derive a general form of the corresponding metric tensor satisfying Einstein’s equations with zero righthand side.

 

Numerical assessment of stratification influence in simple algebraic turbulence model
Uhlíř, V. ; Bodnár, Tomáš ; Caggio, Matteo
This paper presents rst few results obtained using a newly developed test code aimed at validation and crosscomparison of turbulence models to be applied in environmental flows. A simple code based on nite di erence discretization is constructed to solve steady flows of incompresible nonhomogeneous (variable denstity) fluids. For the rst tests a simple algebraic turbulence model was implemented, containing stability function depending on the stratification via the gradient Richardson number. Numerical tests were performed in order to explore the capabilities of the new code and to get some insight into its behavior under di erent stratification. The twodimensional simulations were performed using immersed boundary method for the flow over low smooth hill. The resulting flow fields are compared for selected Richarson numbers ranging from stable up to unstable strati cation conditions.


Numerical validation of a simple immersed boundary solver for branched channels simulations
Lancmanová, A. ; Bodnár, Tomáš ; Keslerová, D.
This contribution reports on an ongoing study of incompressible viscous fluid flow in two dimensional branched channels. A new finite difference solver was developed using a simple implementation of an immersed boundary method to represent the channel geometry. Numerical solutions obtained using this new solver are compared with outputs of an older finite volume code working on classical wall tted structured multiblock grid. Besides of the comparative evaluation of obtained solution, the aim is to verify whether the immersed boundary method is suitable (accurate and e cient enough) for simulations of flow in channels with complicated geometry where the the grid generation might be challenging.

 

Numerical tests of vanishing diffusion stabilization in OldroydB fluid flow simulations
Pires, M. ; Bodnár, Tomáš
This work presents some numerical tests of finite element solution of incompressible OldroydB fluids flows, using different types of numerical stabilization. In this study the diffusive term (Laplacian of extra stress) is added to the tensorial constitutive relation where it is multiplied by a coefficient, that is variable in time. The goal is to make this diffusion coefficient vanish in time, so that the final solution remains unaffected by the added diffusion term. A series of numerical tests was performed for the steady twodimensional OldroydB fluid flow in corrugated channel (tube) to compare different versions of the vanishing stabilization terms and assess their efficiency in enforcing the solution convergence, without affecting the final steady state.


Secondorder model for atmospheric turbulence without critical Richardson number
Caggio, M. ; Schiavon, M. ; Tampieri, F. ; Bodnár, Tomáš
The purpose of this communication is to present a derivation of the nondimensional vertical gradients of the mean wind speed and mean potential temperature expressed in terms of the socalled similarity functions for very stable conditions of the atmosphere where theoretical approaches provide conflicting results (see e.g. Luhar et al. [19]). The result is based on the analysis of the secondorder model equations in the boundary layer approximations in which new heat flux equations are proposed. The model employs a recent closure for the pressuretemperature correlation, avoiding the issue of a critical treshold for the Richardson number.


Programs and Algorithms of Numerical Mathematics 20 : Hejnice, June 2126, 2019 : proceedings of seminar
Chleboun, J. ; Kůs, Pavel ; Přikryl, Petr ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub
This book comprises papers that originated from the invited lectures, survey lectures, short communications, and posters presented at the 20th seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Hejnice, Czech Republic, June 2126, 2020. All the papers have been peerreviewed. The seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EUMATHSIN.cz, Czech Network for Mathematics in Industry. The objective of this series of seminars is to provide a forum for presenting and discussing advanced theoretical as well as practical topics in numerical analysis, computer implementation of algorithms, new approaches to mathematical modeling, and single or multiprocessor applications of computational methods.
