scholarly journals C-N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jinsong Hu ◽  
Bing Hu ◽  
Youcai Xu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Damping Term ◽  
Discrete Energy ◽  
Initial Boundary Problem ◽  
Long Wave

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank-Nicolson nonlinear-implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify the theoretical analysis.

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 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.

scholarly journals A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiaojie Li ◽  
Zhoushun Zheng ◽  
Shuang Wang ◽  
Jiankang Liu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
One Dimensional ◽  
Explicit Finite Difference ◽  
Boltzmann Method

An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmann method. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have been compared with the exact solutions reported in previous studies. TheL2, L∞and Root-Mean-Square (RMS) errors in the solutions show that the scheme is accurate and effective.

scholarly journals FINITE-DIFFERENCE ANALYSIS FOR THE LINEAR THERMOPOROELASTICITY PROBLEM AND ITS NUMERICAL RESOLUTION BY MULTIGRID METHODS

2012 ◽  
Vol 17 (2) ◽  
pp. 227-244 ◽  
Author(s):  
Natalia Boal ◽  
Francisco Jos´e Gaspar ◽  
Francisco Lisbona ◽  
Petr Vabishchevich
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
System Of Equations ◽  
Convergence Properties ◽  
Discrete Energy ◽  
Numerical Resolution ◽  
Multigrid Convergence

This paper deals with the numerical solution of a two-dimensional thermoporoelasticity problem using a finite-difference scheme. Two issues are discussed: stability and convergence in discrete energy norms of the finite-difference scheme are proved, and secondly, a distributive smoother is examined in order to find a robust and efficient multigrid solver for the corresponding system of equations. Numerical experiments confirm the convergence properties of the proposed scheme, as well as fast multigrid convergence.

On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Midpoint Method

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form

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.

COMPARISON OF FINITE‐DIFFERENCE SCHEMES FOR THE GROSS‐PITAEVSKII EQUATION

2009 ◽  
Vol 14 (1) ◽  
pp. 109-126 ◽  
Author(s):  
Vyacheslav A. Trofimov ◽  
Nikolai Peskov
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Laws ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Pitaevskii Equation ◽  
Conservative Scheme ◽  
Conservative Finite Difference Scheme

A conservative finite‐difference scheme for numerical solution of the Gross‐Pitaevskii equation is proposed. The scheme preserves three invariants of the problem: the L 2 norm of the solution, the impulse functional, and the energy functional. The advantages of the scheme are demonstrated via several numerical examples in comparison with some other well‐known and widely used methods. The paper is organized as follows. In Section 2 we consider three main conservation laws of GPE and derive the evolution equations for first and second moments of a solution of GPE. In Section 3 we define the conservative finite‐difference scheme and prove the discrete analogs of conservation laws. The remainder of Section 3 consists of a brief description of other finite‐difference schemes, which will be compared with the conservative scheme. Section 4 presents the results of numerical solutions of three typical problems related to GPE, obtained by different methods. Comparison of the results confirms the advantages of conservative scheme. And finally we summarize our conclusions in Section 5.

A STUDY OF THE SHORT WAVE COMPONENTS IN COMPUTATIONAL ACOUSTICS

1993 ◽  
Vol 01 (01) ◽  
pp. 1-30 ◽  
Author(s):  
CHRISTOPHER K. W. TAM ◽  
JAY C. WEBB ◽  
ZHONG DONG
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Euler Equations ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Long Waves ◽  
Short Waves

It is shown by using a Dispersion-Relation-Preserving [Formula: see text] finite difference scheme that it is feasible to perform direct numerical simulation of acoustic wave propagation problems. The finite difference equations of the [Formula: see text] scheme have essentially the same Fourier-Laplace transforms and hence dispersion relations as the original linearized Euler equations over a broad range of wavenumbers (here referred to as long waves). Thus it is guaranteed that the acoustic waves, the entropy and the vorticity waves computed by the [Formula: see text] scheme are good approximations of those of the exact solutions of Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves of a computation scheme are, therefore, contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration and standard dispersive wave theory. Numerical results of direct simulations of these waves are reported. These waves can be generated by discontinuous initial conditions. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.

Sign in / Sign up

Export Citation Format

Share Document