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Numerical Analysis and Applications

2023 year, number 1

Stability domains of an implicit method for the numerical solution of Abel type integral algebraic equations

O.S. Budnikova1,2, M.N. Botoroeva1,2, G.K. Sokolova1,2
1Irkutsk State University, Irkutsk, Russia
2Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences, Irkutsk, Russia
Keywords: Abel type integral-algebraic equations, Volterra integral equations, k-step methods, stiff problem, stability domains

Abstract >>
This article is devoted to a study of the properties of an implicit method for Abel type integral algebraic equations. An Abel type integral equation with stiff components is used for examining the properties of these methods and the stability domains are constructed. Numerical calculations confirming the results obtained are performed. In this article, a fractional “stiff” problem is proposed to study the stability of the mathematical objects considered.

Formulas for numerical differentiation of functions with large gradients

A.I. Zadorin
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: function of one variable, large gradients, special formula for numerical differentiation, error estimate

Abstract >>
Numerical differentiation of functions with large gradients is investigated. It is assumed that a function contains a component known up to a factor and responsible for the large gradients of the function. Application of classical formulas for calculating derivatives to such functions may lead to significant errors. Special-purpose formulas are studied for numerical differentiation on a uniform grid which are exact for a boundary layer component. Conditions are formulated under which an error estimate of a difference formula for a derivative does not depend on the gradients of the boundary layer component. In the case of an exponential boundary layer, when calculating a derivative of an arbitrarily given order error estimates that are uniform with respect to a small parameter are obtained. The results of numerical experiments are presented.

A local ensemble data assimilation algorithm for nonlinear geophysical models

E.G. Klimova
Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
Keywords: data assimilation, ensemble Kalman filter, particle filter, Gaussian mixture filter

Abstract >>
For optimal estimation of quantities of interest from observational data and a model (optimal filtering problem) in the nonlinear case, a particle method based on a Bayesian approach can be used. A disadvantage of the classical particle filter is that the observations are used only to find the weight coefficients with which the sum of the particles is calculated when determining an estimate. The present article considers an approach to solving the problem of nonlinear filtering which uses a representation of the posterior distribution density of the quantity being estimated as a sum with weights of Gaussian distribution densities. It is well-known from filtration theory that if a distribution density is a sum with weights of Gaussian functions, the optimal estimate will be a sum with weights of estimates calculated by the Kalman filter formulas. The present article proposes a method for solving the problem of nonlinear filtering based on this approach. An ensemble π-algorithm proposed earlier by the author is used to implement the method. The ensemble π-algorithm in this new method is used to obtain an ensemble corresponding to the distribution density at the analysis step. This is a stochastic ensemble Kalman filter which is local as well. Therefore, it can be used in high-dimensional geophysical models.

A posteriori error majorants for numerical solutions of plate bending problems on a Winkler basis

V.G. Korneev
Saint Petersburg State University, St. Petersburg, Russia
Keywords: a posteriori error bounds, singularly perturbed elliptic equations of 4th order, mixed finite element method, lower error bounds

Abstract >>
The paper is devoted to the mixed finite element method for the equation ΔΔυ + κ2υ = ƒ, x ∈ Ω, with boundary conditions υ = ϑυ/ϑν = 0 on ϑΩ, where ν is the normal to the boundary and κ ≥ 0 is an arbitrary constant on each finite element. At κ ≡ 0 residual type a posteriori error bounds for the mixed Ciarlet-Raviart method were derived by several authors at the use of different error norms. The bounds, termed sometimes a posteriori functional error majorants, seem to be less dependent on the constants in the general approximation bounds and are more flexible and adaptable for attaining higher accuracy at practical implementation. In this paper, we present a posteriori functional error majorants for the mixed Ciarlet-Raviart method in the case of κ ≠ 0 and having large jumps. Robustness and sharpness of the bounds are approved by the lower bounds of local efficiency.

Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

I.M. Kulikov, D.A. Karavaev
Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: numerical modeling, computational astrophysics, HLL solver

Abstract >>
A construction of the original HLL method for solving problems of relativistic hydrodynamics by using a piecewise-parabolic reconstruction of the physical variables is described. The resulting numerical method makes it possible to reproduce the numerical solutions with small dissipation at the discontinuities. The method is verified in problems of discontinuity breakdown in one-dimensional and two-dimensional formulation. The accuracy of the numerical scheme is studied in one-dimensional discontinuity breakdown problems. The method is also tested in typical astrophysical problems: interaction of relativistic jets, collision of clouds at relativistic speeds, and supernova explosion.

The source configuration leading to the accumulation of tsunami wave energy around the round island

An.G. Marchuk, E.D. Moskalensky
Institute of Computational Mathematics and Mathematical Geophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: eikonal equation, tsunami, wave front, shallow water equations, wave energy

Abstract >>
A two-dimensional eikonal equation which describes the propagation of tsunami wave fronts is considered. The paper presents the spatial form of an initial source of waves which causes accumulation of wave energy along the coastline of a round island. The theoretical results are confirmed by numerical simulation with a shallow water model of tsunami wave dynamics.

Realization of the adaptation criterion in the grid generation technology for constructions bounded by the surfaces of revolution with parallel axes of revolution

O. V. Ushakova1,2
1N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
2Ural Federal University named after the First President of Russia B.N. Yeltsin, Ekaterinburg, Russia
Keywords: adaptation criterion, technology for grid generation, volumes bounded by the surfaces of revolution with parallel axis of revolution

Abstract >>
A realization of an adaptation criterion in the technology of generation of three-dimensional structured grids designed for the numerical solution of differential equations modeling the vortex processes of multi-component hydrodynamics is described. Earlier the adaptation criterion was realized for volumes of revolution and volumes of revolution deformed by other volumes of revolution. The adaptation criterion is realized within a variational approach for the construction of optimal grids satisfying optimality criteria: closeness of the grid to a uniform and orthogonal one and adaptation to a given function. In the realization of the criterion, the technology is supplemented by a new way of boundary nodes computation and an algorithm for the construction of an admissible set for minimization of a discrete functional formalizing the optimality criteria. Examples of grids adapted to a given function and its first derivatives are given.

Unsteady concentration field of reacting gas in the vicinity of a burning coal particle

S.V. Cherdantsev, P.A. Shlapakov, S.I. Goloskokov, K.S. Lebedev, A.Yu. Erastov
VostNII Scientific Center for Industrial and Environmental Safety in the Mining Industry, Kemerovo, Russia
Keywords: carbon particles, stoichiometric coefficients, spherical coordinates, diffusion coefficient, Arrhenius law, unsteady mass transfer problem, self-similar solution

Abstract >>
The article deals with a boundary value problem of mathematical physics describing the unsteady process of diffusion of a reacting gas to a spherical coal particle located in the atmosphere of a mine. The solution of the boundary value problem is based on self-similar transformations which are a special case of group analysis. Formulas for determining the concentrations of the reacting gas in the vicinity of the coal particle and on its surface are obtained. Graphs of the dependencies of the burn-out time of the coal particle on a number of its parameters are constructed, and the fields of the reacting gas concentration at various stages of combustion of the coal particle are revealed.