Spectral Stability Conditions for an Explicit Three-Level Finite-Difference Scheme for a Multidimensional Transport Equation with Perturbations

Злотник, А. А.; Четверушкин, Б. Н.

doi:10.1134/s0012266121070065

Cited by 3 publications

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References 13 publications

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“…For similar conditions, see [5]- [9] and [11], where 0 < δ < 1, and [10], [16], [17] and [27], where δ = 0.…”

Section: § 1 Introductionmentioning

confidence: 99%

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

Zlotnik,

Chetverushkin

2023

Sb. Math.

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The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.

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“…For similar conditions, see [5]- [9] and [11], where 0 < δ < 1, and [10], [16], [17] and [27], where δ = 0.…”

Section: § 1 Introductionmentioning

confidence: 99%

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

Zlotnik,

Chetverushkin

2023

Sb. Math.

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On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

Злотник

2021

Symmetry

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We deal with 2D and 3D barotropic gas dynamics system of equations with two viscous regularizations: so-called quasi-gas dynamics (QGD) and quasi-hydrodynamics (QHD) ones. The system is linearized on a constant solution with any velocity, and an explicit two-level in time and symmetric three-point in each spatial direction finite-difference scheme on the uniform rectangular mesh is considered for the linearized system. We study L2-dissipativity of solutions to the Cauchy problem for this scheme by the spectral method and present a criterion in the form of a matrix inequality containing symbols of symmetric matrices of convective and regularizing terms. Analyzing these inequality and matrices, we also derive explicit sufficient conditions and necessary conditions in the Courant-type form which are rather close to each other. For the QHD regularization, such conditions are derived for the first time in 2D and 3D cases, whereas, for the QGD regularization, they improve those that have recently been obtained. Explicit formulas for a scheme parameter that guarantee taking the maximal time step are given for these conditions. An important moment is a new choice of an “average” spatial mesh step ensuring the independence of the conditions from the ratios of the spatial mesh steps and, for the QGD regularization, from the Mach number as well.

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О Свойствах И Погрешности Параболического И Гиперболического 2-Го Порядка Возмущений Симметричной Гиперболической Системы 1-Го Порядка

Злотник¹,

Четверушкин²

2023

Математический сборник

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Изучаются задачи Коши для многомерной симметричной линейной гиперболической системы уравнений 1-го порядка с переменными коэффициентами и ее сингулярных возмущений - сильно параболической и гиперболической 2-го порядка систем уравнений с малым параметром $\tau>0$ при вторых производных по $x$ и $t$. Доказываются существование и единственность слабых решений всех трех систем и равномерные по $\tau$ оценки решений систем с возмущениями. Даются оценки разности решений исходной системы и систем с возмущениями, в том числе в норме $C(0,T;L^2(\mathbb{R}^n))$ порядка $O(\tau^{\alpha/2})$ при начальной функции $\mathbf w_0$ из пространств Соболева $H^\alpha(\mathbb{R}^n)$ для $\alpha=1,2$ и пространств Никольского $H_2^{\alpha}(\mathbb{R}^n)$ для $0<\alpha<2$, $\alpha\neq 1$ и соответствующих условиях на свободный член системы 1-го порядка. При $\alpha=1/2$ охватывается широкий класс разрывных $\mathbf w_0$. Выводятся также оценки производных любого порядка по $x$ как решений, так и их разностей порядка $O(\tau^{\alpha/2})$. Указывается приложение результатов к линеаризованной на постоянном решении системе уравнений газовой динамики 1-го порядка и ее возмущениям - линеаризованным параболической и гиперболической 2-го порядка квазигазодинамическим системам уравнений. Библиография: 34 названия.

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Spectral Stability Conditions for an Explicit Three-Level Finite-Difference Scheme for a Multidimensional Transport Equation with Perturbations

Cited by 3 publications

References 13 publications

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

On Conditions for L2-Dissipativity of an Explicit Finite-Difference Scheme for Linearized 2D and 3D Barotropic Gas Dynamics System of Equations with Regularizations

О Свойствах И Погрешности Параболического И Гиперболического 2-Го Порядка Возмущений Симметричной Гиперболической Системы 1-Го Порядка

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