On a hyperbolic perturbation of a parabolic initial–boundary value problem

Четверушкин, Б. Н.; Злотник, А. А.

doi:10.1016/j.aml.2018.03.027

Cited by 16 publications

(4 citation statements)

References 8 publications

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“…Note that second-order hyperbolic perturbations of parabolic problems have also been intensively studied in the literature, including estimates for the difference of their solutions obtained in [19]- [22]. § 2.…”

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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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 papers [4,5] establish a number of mathematical properties of the quasi-gasdynamic system, including timeuniform estimates for the corresponding system linearized on a constant solution. A second-order hyperbolic perturbation with the parameter τ of the parabolic initial-boundary value problem without convective terms [6,7] and with them [8] was also studied, and the stability of the corresponding three-level weighted and two-level vector numerical methods was analyzed [9]. The time-uniform stability of implicit three-level weighted and two-level vector difference schemes for the linearized quasi-gasdynamic system was proved in [10].…”

Section: Introductionmentioning

confidence: 99%

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

Злотник

Четверушкин

2021

Diff Equat

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We study difference schemes associated with a simplified linearized multidimensional hyperbolic quasi-gasdynamic system of differential equations. It is shown that an explicit two-level vector difference scheme with flux relaxation for a second-order hyperbolic equation with variable coefficients that is a perturbation of the transport equation with a parameter multiplying the highest derivatives can be reduced to an explicit three-level difference scheme. In the case of constant coefficients, the spectral condition for the time-uniform stability of this explicit three-level difference scheme is analyzed, and both sufficient and necessary conditions for this condition to hold are derived, in particular, in the form of Courant type conditions on the ratio of temporal and spatial steps.

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“…The QGD model has an advantage that it guarantees the smoothing of the solution at the free path distance. The QGD equations are extensively described in the literature [6,7,8,9,10,35].…”

Section: Introductionmentioning

confidence: 99%

“…In literature, this model has also been used to regularize purely parabolic equations by adding a hyperbolicity. This regularization has been employed in designing efficient time stepping algorithms [7,9,10].…”

Section: Introductionmentioning

confidence: 99%

Computational multiscale methods for quasi-gas dynamic equations

Chetverushkin,

Chung,

Efendiev

et al. 2020

Preprint

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In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been focused on problems with constant coefficients. In this paper, we investigate the QGD model in multiscale media, which can be used in porous media applications. This multiscale problem is interesting from a multiscale methodology point of view as the model problem has a hyperbolic multiscale term, and designing multiscale methods for hyperbolic equations is challenging. In the paper, we apply the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) combined with the leapfrog scheme in time to solve this problem. The CEM-GMsFEM provides a flexible and systematical framework to construct crucial multiscale basis functions for approximating the solution to the problem with reduced computational cost. With this approach of spatial discretization, we establish the stability of the fully discretized scheme under a relaxed version of the so-called CFL condition. Complete convergence analysis of the proposed method is presented. Numerical results are provided to illustrate and verify the theoretical findings.

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On a hyperbolic perturbation of a parabolic initial–boundary value problem

Cited by 16 publications

References 8 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

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

Computational multiscale methods for quasi-gas dynamic equations

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