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 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 Conservative Linear Difference Scheme for Rosenau-KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Theoretical Results

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

scholarly journals An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Maobo Zheng ◽  
Jun Zhou
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Energy ◽  
The Difference ◽  
Initial Boundary Value ◽  
Theoretical Results

An average linear finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-KdV equation is proposed. The existence, uniqueness, and conservation for energy of the difference solution are proved by the discrete energy norm method. It is shown that the finite difference scheme is 2nd-order convergent and unconditionally stable. Numerical experiments verify that the theoretical results are right and the numerical method is efficient and reliable.

Numerical Solution Of A Hyperbolic Transmission Problem

2008 ◽  
Vol 8 (4) ◽  
pp. 374-385 ◽  
Author(s):  
B.S. JOVANOVIC ◽  
L.G. VULKOV
Keyword(s):  
Finite Difference ◽  
Convergence Rate ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Hyperbolic Equation ◽  
Finite Difference Scheme ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
One Dimensional ◽  
Initial Boundary Value

AbstractIn this paper we investigate an initial boundary value problem for a one-dimensional hyperbolic equation in two disconnected intervals. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate has been obtained.

Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jinye Shen ◽  
Martin Stynes ◽  
Zhi-Zhong Sun
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Mesh Size ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Initial Time ◽  
Finite Difference Schemes

Abstract A time-fractional initial-boundary value problem of wave type is considered, where the spatial domain is ( 0 , 1 ) d (0,1)^{d} for some d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} . Regularity of the solution 𝑢 is discussed in detail. Typical solutions have a weak singularity at the initial time t = 0 t=0 : while 𝑢 and u t u_{t} are continuous at t = 0 t=0 , the second-order derivative u t ⁢ t u_{tt} blows up at t = 0 t=0 . To solve the problem numerically, a finite difference scheme is used on a mesh that is graded in time and uniform in space with the same mesh size ℎ in each coordinate direction. This scheme is generated through order reduction: one rewrites the differential equation as a system of two equations using the new variable v := u t v:=u_{t} ; then one uses a modified L1 scheme of Crank–Nicolson type for the driving equation. A fast variant of this finite difference scheme is also considered, using a sum-of-exponentials (SOE) approximation for the kernel function in the Caputo derivative. The stability and convergence of both difference schemes are analysed in detail. At each time level, the system of linear equations generated by the difference schemes is solved by a fast Poisson solver, thereby taking advantage of the fast difference scheme. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of both numerical methods.

scholarly journals Additive difference scheme for two-dimensional fractional in time diffusion equation

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 217-226 ◽  
Author(s):  
Sandra Hodzic-Ivanovic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

An additive finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

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 Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

2016 ◽  
Vol 99 (113) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksandra Delic ◽  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Nonhomogeneous Media ◽  
Subdiffusion Equation ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

Finite Difference Approximation of Fractional Wave Equation with Concentrated Capacity

2017 ◽  
Vol 17 (1) ◽  
pp. 33-49 ◽  
Author(s):  
Aleksandra Delić ◽  
Boško S. Jovanović
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Generalized Solutions ◽  
Difference Approximation ◽  
Dirac Delta ◽  
Fractional Wave Equation ◽  
Initial Boundary Value

AbstractWe consider the time fractional wave equation with coefficient which contains the Dirac delta distribution. The existence of generalized solutions of this initial-boundary value problem is proved. An implicit finite difference scheme approximating the problem is developed and its stability is proved. Estimates for the rate of convergence in special discrete energetic Sobolev norms are obtained. A numerical example confirms the theoretical results.

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