scholarly journals On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions

2014 ◽  
Vol 19 (2) ◽  
pp. 225-240 ◽  
Author(s):  
Justina Jachimavičienė ◽  
Mifodijus Sapagovas ◽  
Artūras Štikonas ◽  
Olga Štikonienė
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlocal Conditions ◽  
Finite Difference Schemes ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
Explicit Finite Difference ◽  
The Stability

A new explicit conditionally consistent finite difference scheme for one-dimensional third-order linear pseudoparabolic equation with nonlocal conditions is constructed. The stability of the finite difference scheme is investigated by analysing a nonlinear eigenvalue problem. The stability conditions are stated and stability regions are described. Some numerical experiments are presented in order to validate theoretical results.

scholarly journals Explicit Finite-Difference Scheme for the Numerical Solution of the Model Equation of Nonlinear Hereditary Oscillator with Variable-Order Fractional Derivatives

2016 ◽  
Vol 26 (3) ◽  
pp. 429-435 ◽  
Author(s):  
Roman I. Parovik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Fractional Derivatives ◽  
Stability And Convergence ◽  
Variable Order ◽  
The Difference ◽  
Explicit Finite Difference ◽  
The Stability ◽  
Variable Order Fractional Derivatives

Abstract The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is stable and converges to the exact solution.

The Finite Difference Scheme for 3d Mathematical Modeling of a Wood Drying Process

2001 ◽  
Vol 1 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Raimondas Čiegis ◽  
Vadimas Starikovičius
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Numerical Experiments ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
The Stability ◽  
Robin Boundary

AbstractThis work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

scholarly journals On the stability of a finite difference scheme with two weights for wave equation with nonlocal conditions

2014 ◽  
Vol 55 ◽  
pp. 22-27
Author(s):  
Jurij Novickij ◽  
Artūras Štikonas
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability Region ◽  
Nonlocal Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Nonlocal Integral Boundary Conditions ◽  
The Stability ◽  
Param Eters

We consider the stability of a finite difference scheme with two weight param-eters for a hyperbolic equation with nonlocal integral boundary conditions. We obtain stability region in the complex plane by investigating the characteristic equation of a difference scheme using the root criterion. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant MIP-047/2014.

THE FINITE DIFFERENCE SCHEME FOR MATHEMATICAL MODELING OF WOOD DRYING PROCESS

2001 ◽  
Vol 6 (1) ◽  
pp. 48-57 ◽  
Author(s):  
R. Čiegis ◽  
V. Starikovičius
Keyword(s):  
Mathematical Modeling ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Drying Process ◽  
Moisture Movement ◽  
Different Types ◽  
The Stability

This work discusses issues on the design of finite difference schemes for modeling the moisture movement process in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for different types of boundary conditions.

Removing the stability limit of the explicit finite-difference scheme with eigenvalue perturbation

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. A93-A98 ◽  
Author(s):  
Yingjie Gao ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Unit Circle ◽  
Finite Difference Scheme ◽  
Stability Limit ◽  
Time Step ◽  
Dispersion Error ◽  
Time Dispersion ◽  
Explicit Finite Difference ◽  
The Stability

The explicit finite-difference scheme is popular for solving the wave equation in the field of seismic exploration due to its simplicity in numerical implementation. However, its maximum time step is strictly restricted by the Courant-Friedrichs-Lewy (CFL) stability limit, which leads to a heavy computational burden in the presence of small-scale structures and high-velocity targets. We remove the CFL stability limit of the explicit finite-difference scheme using the eigenvalue perturbation, which allows us to use a much larger time step beyond the CFL stability limit. For a given time step that is within the CFL stability limit, the eigenvalues of the update matrix would be distributed along the unit circle; otherwise, some eigenvalues would be distributed outside of the unit circle, which introduces unstable phenomena. The eigenvalue perturbation can normalize the unstable eigenvalues and guarantee the stability of the update matrix by using an arbitrary time step. The update matrix can be preprocessed before the numerical simulation, thus retaining the computational efficiency well. We further incorporate the forward time-dispersion transform (FTDT) and the inverse time-dispersion transform (ITDT) to reduce the time-dispersion error caused by using an unusually large time step. Our numerical experiments indicate that the combination of the eigenvalue perturbation, the FTDT method, and the ITDT method can simulate highly accurate waveforms when applying a time step beyond the CFL stability limit. The time step can be extended even toward the Nyquist limit. This means that we could save many iteration steps without suffering from time-dispersion error and stability problems.

scholarly journals ON CONSTRUCTION AND ANALYSIS OF FINITE DIFFERENCE SCHEMES FOR PSEUDOPARABOLIC PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

2014 ◽  
Vol 19 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Raimondas Čiegis ◽  
Natalija Tumanova
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Two Dimensional ◽  
The Stability

In this paper the one- and two-dimensional pseudoparabolic equations with nonlocal boundary conditions are approximated by the Euler finite difference scheme. In the case of classical boundary conditions the stability of all schemes is investigated by the spectral method. Stability regions of finite difference schemes approximating pseudoparabolic problem are compared with the stability regions of the classical discrete parabolic problem. These results are generalized for problems with nonlocal boundary conditions if a matrix of the finite difference scheme can be diagonalized. For the two-dimensional problem an efficient algorithm is constructed, which is based on the combination of the FFT method and the factorization algorithm. General stability results, known for the three level finite difference schemes, are applied to investigate the stability of some explicit approximations of the two-dimensional pseudoparabolic problem with classical boundary conditions. A connection between the energy method stability conditions and the spectrum Hurwitz stability criterion is shown. The obtained results can be applied for pseudoparabolic problems with nonlocal boundary conditions, if a matrix of the finite difference scheme can be diagonalized.

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