Exact Difference Schemes for Hyperbolic Equations

2007 ◽  
Vol 7 (4) ◽  
pp. 341-364 ◽  
Author(s):  
P. Matus ◽  
A. Kołodyńska
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Difference Scheme ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Cauchy Problem ◽  
Exact Difference ◽  
Constant Coefficients ◽  
Initial Boundary Value

AbstractIn the present paper, an exact difference scheme for the initial boundary- value problem of the third kind for an inhomogeneous hyperbolic equation of the second order with constant coefficients has been constructed on ordinary rectangular grids with constant space and time steps, where the Courant number γ=1. Later we proved a priori estimates of the stability in energy norm. For a quasi-linear wave equation on the moving characteristic grid a difference scheme has been constructed, which has the second order of approximation for the initial boundary-value problem and is exact for the Cauchy problem. The computational results for smooth functions and for a weak solution confirm the high accuracy of the introduced algorithm. We have also constructed exact difference schemes for the Cauchy problem for a system of two hyperbolic equations of the first order with constant coefficients on grids with constant space and time steps. Stability in energy norm for one of the constructed schemes has been proved. Using a method analogous to that used for the nonlinear wave equation a difference scheme for a nonlinear gas dynamic system has been constructed.

scholarly journals A linear finite difference scheme for the generalized dissipative symetric regularized long wave equation with damping

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 719-726 ◽  
Author(s):  
Xi Wang ◽  
Jin-Song Hu ◽  
Hong Zhang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Initial Boundary Value

In this paper, we study and analyze a three-level linear finite difference scheme for the initial boundary value problem of the symmetric regularized long wave equation with damping. The proposed scheme has the second accuracy both for the spatial and temporal discretization. The convergence and stability of the numerical solutions are proved by the mathematical induction and the discrete functional analysis. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm.

scholarly journals Note on the Uniqueness Proprety of Weak Solutions of Parabolic Equations

1967 ◽  
Vol 29 ◽  
pp. 45-49 ◽  
Author(s):  
Tadashi Kuroda
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Property ◽  
Uniqueness Of Solutions ◽  
Measurable Coefficients ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Aronson proved, in his paper [1], the existence and the uniqueness property of weak solutions of the initial boundary value problem for parabolic equations of second order with measurable coefficients. On the uniqueness of solutions of the Cauchy problem for such equations he also gave some interesting results in [2].

scholarly journals Stability with respect to coefficients of solution of difference schemes approximating initial boundary-value problems for semi-linear hyperbolic equations

2020 ◽  
Vol 64 (2) ◽  
pp. 135-143
Author(s):  
P. P. Matus ◽  
S. V. Lemeshevsky
Keyword(s):  
Difference Scheme ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Differential Problem ◽  
Domain Of Existence ◽  
The Stability ◽  
The One ◽  
Initial Boundary Value

The stability with respect to coefficients of solution of a difference scheme approximating the initial boundary-value problem for the one-dimensional semi-linear hyperbolic equation is studied. The estimates of the solutions of both differential and difference problems are obtained. In the domain of existence of the solution, the estimates for perturbation of the solution of a difference scheme with respect to perturbation of the coefficients of the equation are obtained. These estimates are consistent with the estimates for the differential problem. In all cases, the method of energy inequalities, the Bihari inequality and its mesh analogue are used.

scholarly journals PRACTICAL ERROR ANALYSIS FOR THE THREE-LEVEL BILINEAR FEM AND FINITE-DIFFERENCE SCHEME FOR THE 1D WAVE EQUATION WITH NON-SMOOTH DATA

2018 ◽  
Vol 23 (3) ◽  
pp. 359-378
Author(s):  
Alexander Zlotnik ◽  
Olga Kireeva
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Free Term ◽  
The Rich ◽  
Initial Boundary Value

We deal with the standard three-level bilinear FEM and finite-difference scheme with a weight to solve the initial-boundary value problem for the 1D wave equation. We consider the rich collection of initial data and the free term which are the Dirac δ-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the L2, L1, energy and uniform norms as the mesh re_nes and compare results with known theoretical error bounds.

scholarly journals Numerical Solution of Stochastic Hyperbolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Necmettin Aggez ◽  
Maral Ashyralyyewa
Keyword(s):  
Difference Scheme ◽  
Numerical Solution ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Estimate ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
The Difference ◽  
Initial Boundary Value

A two-step difference scheme for the numerical solution of the initial-boundary value problem for stochastic hyperbolic equations is presented. The convergence estimate for the solution of the difference scheme is established. In applications, the convergence estimates for the solution of the difference scheme are obtained for different initialboundary value problems. The theoretical statements for the solution of this difference scheme are supported by numerical examples.

scholarly journals Convergence to diffusion waves for solutions of 1D Keller-Segel model

2021 ◽  
Author(s):  
Fengling Liu ◽  
Nangao Zhang ◽  
Changjiang Zhu
Keyword(s):  
Boundary Condition ◽  
Cauchy Problem ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Initial Boundary Value

In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initial-boundary value problem) of one-dimensional Keller-Segel model. For the Cauchy problem, we prove that the solutions time-asymptotically converge to the nonlinear diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which is derived by Darcy’s law, as in [11, 28]. For the initial-boundary value problem, we consider two cases: Dirichlet boundary condition and null-Neumann boundary condition on (u, ρ). In the case of Dirichlet boundary condition, similar to the Cauchy problem, the asymptotic profile is still the self similar solution of the corresponding parabolic equation, which is derived by Darcy’s law, thus we only need to deal with boundary effect. In the case of null-Neumann boundary condition, the global existence and asymptotic behavior of solutions near constant steady states are established. The proof is based on the elementary energy method and some delicate analysis of the corresponding asymptotic profiles.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

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