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A Compact 4th-Order Scheme for the General Acoustic Wave Equation
Lobachevskii Journal of Mathematics. 2026. Vol. 47. No. 5. P. 2446–2460.
We deal with the general multidimensional acoustic wave equation, where both the speed of sound and density of a medium are variable, that is physically relevant. For the first time, we develop a new compact scheme for this equation which is three-level in time, three-point in each spatial direction and has the fourth truncation order $\mathcal{O}(|h|^4+h_t^4)$. In addition, the scheme is semi-explicit in time and easily implementable. It non-trivially generalizes a semi-explicit vector compact scheme constructed and studied recently in the much simpler case of the constant density. Although other high-order numerical methods for the general acoustic wave equation were known, the construction of a compact high-order scheme remained an unsolved problem for a long time. Results of computations confirm the high precision of the scheme and its fourth error order not only in the mesh $C$ norm but in the mesh $C^1$ norm as well that is of value in applications.
Keywords: practical error analysisacoustic wave equationcompact fourth-order schemevariable speed of sound and densitysemi-explicit three-level scheme
Publication based on the results of:
Bernardin C., Chhaibi R., Najnudel J. et al., Probability Theory and Related Fields 2026 Vol. 195 P. 1823–1875
We study the celebrated Shiryaev-Wonham filter (Wonham, W.M., in J. Soc. Ind. Appl. Math. 347–369, 1964) in its historical setup, where the hidden Markov jump process has two states. We are interested in the weak noise regime for the observation equation. Interestingly, this becomes a strong noise regime for the filtering equations. Earlier results of ...
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Abdulkhaev K., Shirokov D., Advances in Applied Clifford Algebras 2026 Vol. 36 P. 1–21
In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension n = 7. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces ...
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The dynamics of opinion in a group are of interest for a number of practical purposes. In particular, consensus helps and polarization of opinions hinders cohesive teamwork. Existing approaches for modeling opinion dynamics mostly do not take into account the dynamism of social relations in a group. This paper proposes an algorithm and a program ...
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Decision trees and their ensemble extensions, such as random forests, are widely used as classification models due to their simplicity and interpretability. However, in many real-world tasks where class labels overlap in the feature space, standard decision trees rely on hard splits that create fragile decision boundaries. In these regions, small perturbations in the input ...
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Medvedev G., Alexandrov Artem, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2026 Vol. 114 Article 044102
Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include ...
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Pochinka O., Baranov D., Nozdrinova E., Теоретическая и математическая физика 2026 Т. 229 № 1 С. 3–14
The Birman–Williams problem on describing the planetary link of a fibered knot K in S^3 has been partially solved. Using Nielsen's theory for the classification of periodic surface homeomorphisms and its close relationship with the theory of gradient-like diffeomorphisms, it is proved that the planetary link of the trefoil (the unique periodic fibered knot of genus ...
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Prokhorov A., Burda M., Journal of Computational and Graphical Statistics 2026 P. 1–13
Bayesian nonparametric density estimation procedures are typically based on single-scale priors, such as Dirichlet process mixtures. Alternative multiscale density priors built on decision trees have many well-known advantages, including the ability to characterize abrupt local changes and to provide an estimate with a desired level of resolution. Despite their theoretical appeal, multiscale methods have typically ...
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Zlotnik A., Математические заметки 2026 Т. 120 № 6 С. 1005–1009
Численным методам решения систем газодинамических уравнений посвящена обширная литература. Ранее было разработано и успешно апробировано специальное семейство симметричных по пространству консервативных разностных методов, основанных на предварительной кинетической, точнее, квазигазодинамической (КГД), регуляризации этих уравнений. Актуальной задачей является построение численных методов, которые обладают не только свойством консервативности по массе, импульсу и полной энергии, но и удовлетворяют условиям энтропийной ...
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Vyugin I. V., Sashadhar D., Algebra and Number Theory 2026 P. 1–10
We study the K-Fibonacci sequence Fp modulo prime p. Cardinalities of sets |Fp+Fp| and |Fp⋅Fp| are estimated. We present the method of estimating doubling constant of some m-dimensional recurrent sets in Fp. ...
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Kuksin S., Dynamical Systems 2026
We study the mixing properties of discrete-time and continuous-time dissipative dynamical systems driven by bounded mixing random forces. The continuous-time systems are
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Zlotnik A., Journal of Complexity 2025 Vol. 91 Article 101973
A compact three-level fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order $\mathcal{O}(h^{4(\lambda-1)/5})$ are proved in the mesh energy norm in terms of data, for two initial functions from the Sobolev and Nikolskii spaces with the smoothness orders $\lambda$ and $\lambda-1$ and the free term with a ...
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Zlotnik A., Lomonosov T., Russian Journal on Numerical Analysis and Mathematical Modelling 2025 Vol. 40 No. 1 P. 71–88
We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove conditional stability and derive 4th order error bound in the enlarged energy norm. It exploits additional sought functions which approximate 2nd order non-mixed spatial ...
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Zlotnik A., Lomonosov T., Applied Numerical Mathematics 2024 Vol. 195 P. 54–74
We study a three-level explicit in time higher-order vector compact scheme, with additional n sought functions approximating second order non-mixed spatial derivatives of the solution, for an initial-boundary value problem for the n-dimensional wave equation and acoustic wave equation, with the variable speed of sound, n⩾1. We also approximate the solution at the first time level in a ...
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We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The alternative ...
Added: December 9, 2020
Zlotnik A., Kireeva O., / Series arXiv "math". 2020. No. arXiv:2011.14104v2[math.NA].
We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The ...
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Zlotnik A., Lomonosov T., Mathematical Modelling and Analysis 2019 Vol. 24 No. 2 P. 179–194
An entropy dissipative spatial discretization has recently been constructed for the multidimensional gas dynamics equations based on their preliminary parabolic quasi-gasdynamic (QGD) regularization. In this paper, an explicit finite-difference scheme with such a discretization is verified on several versions of the 1D Riemann problem, both well-known in the literature and new. The scheme is compared with the ...
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Zlotnik A., Kireeva O., Mathematical Modelling and Analysis 2018 Vol. 23 No. 3 P. 359–378
We deal with the standard three-level bilinear FEM and finite-difference scheme to solve the initial-boundary value problem for the 1D wave equation. We consider initial data and the free term which are the Dirac delta-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the $L^2$, $L^1$, ...
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Zlotnik Alexander, Zlotnik Ilya, Computational Methods in Applied Mathematics 2015 Vol. 15 No. 2 P. 233–245
We consider the Cauchy problem for the 1D generalized Schrὅdinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order ...
Added: March 3, 2015