scholarly journals On stable high order difference schemes for hyperbolic problems with the Neumann boundary conditions

2019 ◽  
Vol 9 (1) ◽  
pp. 60-72 ◽  
Author(s):  
Ozgur Yildirim
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Order Difference

In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed.

scholarly journals Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations

2005 ◽  
Vol 2005 (2) ◽  
pp. 183-213 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskii
Keyword(s):  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
High Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Definite Operator ◽  
Accuracy Difference ◽  
High Order Of Accuracy ◽  
The Stability

We consider the abstract Cauchy problem for differential equation of the hyperbolic typev″(t)+Av(t)=f(t)(0≤t≤T),v(0)=v0,v′(0)=v′0in an arbitrary Hilbert spaceHwith the selfadjoint positive definite operatorA. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on the three points for the numerical solutions of this problem are presented. The stability estimates for the solutions of these difference schemes are established. In applications, the stability estimates for the solutions of the high order of accuracy difference schemes of the mixed-type boundary value problems for hyperbolic equations are obtained.

scholarly journals Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Necmettin Aggez
Keyword(s):  
Space Variable ◽  
Hyperbolic Equations ◽  
Nonlocal Boundary ◽  
Integral Condition ◽  
Variable Coefficients ◽  
Difference Schemes ◽  
Order Difference ◽  
One Dimensional ◽  
Nonlocal Integral Condition ◽  
Multidimensional Hyperbolic Equations

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-dimensional hyperbolic equations are supported by numerical examples.

On numerical stability analysis of double-diffusive convection in confined enclosures

2001 ◽  
Vol 433 ◽  
pp. 209-250 ◽  
Author(s):  
M. MAMOU ◽  
P. VASSEUR ◽  
M. HASNAOUI
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Travelling Waves ◽  
Neumann Boundary ◽  
Finite Amplitude ◽  
Parallel Flow ◽  
Neumann Boundary Conditions ◽  
Thermosolutal Convection ◽  
Quiescent State

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

scholarly journals A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ozgur Yildirim
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Order Of Accuracy ◽  
Hyperbolic Problem ◽  
Definite Operator ◽  
The Stability

The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceHwith the self-adjoint positive definite operatorA. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

On the Treatment of Neumann Boundary Conditions in Collocation-Based Meshless Methods

2013 ◽  
Vol 423-426 ◽  
pp. 1757-1762
Author(s):  
Xiang Dong Zhang ◽  
Lei Wang ◽  
Da Wei Teng
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Shape Functions ◽  
Numerical Accuracy ◽  
The Poor ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Methods Of Treatment

The existence of Neumann boundary is a major cause of the poor accuracy and instability of collocation-based methods. Taking a Poisson equation with Neumann boundary condition as the model, the present paper studies the effects of two different radial point interpolation shape functions and their parameters on the accuracy of numerical solutions of the equation. We also study the effects of methods including fictious point method, nodes densification method and Hermite collocation method on the improvement of numerical accuracy. By comparison of analytic and numerical solutions computed using a program developed during research, we obtain parameters of shape functions and methods of treatment of Neumann boundary conditions that can be adopted to give better numerical accuracy.

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