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

2021 ◽  
Vol 57 (7) ◽  
pp. 891-900
Author(s):  
A. A. Zlotnik ◽  
B. N. Chetverushkin
Keyword(s):  
Difference Scheme ◽  
Transport Equation ◽  
Variable Coefficients ◽  
Spectral Condition ◽  
Level Difference ◽  
Vector Difference ◽  
Constant Coefficients ◽  
Sufficient And Necessary Conditions ◽  
Order Hyperbolic Equation ◽  
Flux Relaxation

Abstract 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.

Method of corrections by higher order differences for elliptic equations with variable coefficients

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Givi Berikelashvili ◽  
Bidzina Midodashvili
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Convergence Rate ◽  
Difference Scheme ◽  
Elliptic Equations ◽  
Variable Coefficients ◽  
Order Accuracy ◽  
Corrected Scheme

AbstractWe consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

2018 ◽  
Vol 106 (1) ◽  
pp. 19-30
Author(s):  
MICHAEL GIL’
Keyword(s):  
Differential Operators ◽  
Normal Operator ◽  
Variable Coefficients ◽  
Closed Convex Hull ◽  
Constant Matrix ◽  
Norm Estimates ◽  
Matrix Coefficients ◽  
Matrix Differential Operators ◽  
Constant Coefficients ◽  
Matrix Differential

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

scholarly journals Monte-Carlo for solving large linear systems of ordinary differential equations

2021 ◽  
Vol 8 (1) ◽  
pp. 37-48
Author(s):  
Sergey M. Ermakov ◽  
◽  
Maxim G. Smilovitskiy ◽  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Differential Equations ◽  
Integral Equations ◽  
Virtual Machines ◽  
Variable Coefficients ◽  
Fredholm Type ◽  
Type Integral ◽  
Constant Coefficients ◽  
Linear Differential

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

scholarly journals Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

2006 ◽  
Vol 6 (2) ◽  
pp. 154-177 ◽  
Author(s):  
E. Emmrich ◽  
R.D. Grigorieff
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Elliptic Boundary ◽  
Special Kind ◽  
Variable Coefficients ◽  
Second Order ◽  
Nonuniform Grids ◽  
Close Relationship

AbstractIn this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.

Sign in / Sign up

Export Citation Format

Share Document