A FOURTH ORDER ENERGY STABLE FINITE DIFFERENCE SCHEME FOR A TIMOSHENKO BEAM EQUATIONS WITH LOCALLY DISTRIBUTED FEEDBACK

2021 ◽  
Vol 10 (10) ◽  
pp. 3283-3296
Author(s):  
S. Rechdaoui ◽  
A. Taakili
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Timoshenko Beam ◽  
Continuous Model ◽  
Discrete Analogue ◽  
Distributed Feedback ◽  
Energy Relation ◽  
Order Energy ◽  
Beam Equations ◽  
Spatial Derivatives

This work deals with the numerical solution of a control problem governed by the Timoshenko beam equations with locally distributed feedback. We apply a fourth-order Compact Finite Difference (CFD) approximation for the discretizing spatial derivatives and a Forward second order method for the resulting linear system of ordinary differential equations. Using the energy method, we derive energy relation for the continuous model, and design numerical scheme that preserve a discrete analogue of the energy relation. Numerical results show that the CFD approximation of fourth order give an efficient method for solving the Timoshenko beam equations.

A highly stable explicit scheme for a fourth-order nonlinear diffusion filter

2020 ◽  
Vol 11 (04) ◽  
pp. 2050030
Author(s):  
Mahipal Jetta
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Fourth Order ◽  
Nonlinear Diffusion ◽  
Splitting Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Diffusion Filter ◽  
Spatial Derivatives

The standard finite difference scheme (forward difference approximation for time derivative and central difference approximations for spatial derivatives) for fourth-order nonlinear diffusion filter allows very small time-step size to obtain stable results. The alternating directional implicit (ADI) splitting scheme such as Douglas method is highly stable but compromises accuracy for a relatively larger time-step size. In this paper, we develop [Formula: see text] stencils for the approximation of second-order spatial derivatives based on the finite pointset method. We then make use of these stencils for approximating the fourth-order partial differential equation. We show that the proposed scheme allows relatively bigger time-step size than the standard finite difference scheme, without compromising on the quality of the filtered image. Further, we demonstrate through numerical simulations that the proposed scheme is more efficient, in obtaining quality filtered image, than an ADI splitting scheme.

scholarly journals NONSTANDARD FINITE DIFFERENCE SCHEMES FOR SOLVING A MODIFIED EPIDEMIOLOGICAL MODEL FOR COMPUTER VIRUSES

2018 ◽  
Vol 34 (2) ◽  
pp. 171-185 ◽  
Author(s):  
Tuan Manh Hoang ◽  
A Quang Dang ◽  
Long Quang Dang
Keyword(s):  
Finite Difference ◽  
Numerical Simulations ◽  
Fourth Order ◽  
Continuous Model ◽  
Propagation Model ◽  
Finite Difference Schemes ◽  
Virus Propagation ◽  
Order Of Accuracy ◽  
Essential Properties ◽  
Nonstandard Finite Difference

In this paper we construct two families of nonstandard finite difference (NSFD) schemes preserving the essential properties of a computer virus propagation model, such as positivity, boundedness and stability. The first family of NSFD schemes is constructed based on the nonlocal discretization and has first order of accuracy, while the second one is based on the combination of a classical Runge-Kutta method and selection of a nonstandard denominator function and it is of fourth order of accuracy. The theoretical study of these families of NSFD schemes is performed with support of numerical simulations. The numerical simulations confirm the accuracy and the efficiency of the fourth order NSFD schemes. They hint that the disease-free equilibrium point is not only locally stable but also globally stable, and then this fact is proved theoretically. The experimental results also show that the global stability of the continuous model is preserved.

scholarly journals Boundary closures for fourth-order energy stable weighted essentially non-oscillatory finite-difference schemes

2011 ◽  
Vol 230 (10) ◽  
pp. 3727-3752 ◽  
Author(s):  
Travis C. Fisher ◽  
Mark H. Carpenter ◽  
Nail K. Yamaleev ◽  
Steven H. Frankel
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Order Energy ◽  
Energy Stable

A stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space for the scalar wave equation

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. T37-T42 ◽  
Author(s):  
Jing-Bo Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Fourth Order ◽  
Second Order ◽  
High Order ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
General Order ◽  
Second Order In Time ◽  
Spatial Derivatives

Based on the formula for stability of finite-difference methods with second-order in time and general-order in space for the scalar wave equation, I obtain a stability formula for Lax-Wendroff methods with fourth-order in time and general-order in space. Unlike the formula for methods with second-order in time, this formula depends on two parameters: one parameter is related to the weights for approximations of second spatial derivatives; the other parameter is related to the weights for approximations of fourth spatial derivatives. When discretizing the mixed derivatives properly, the formula can be generalized to the case where the spacings in different directions are different. This formula can be useful in high-accuracy seismic modeling using the scalar wave equation on rectangular grids, which involves both high-order spatial discretizations and high-order temporal approximations. I also prove the instability of methods obtained by applying high-order finite-difference approximations directly to the second temporal derivative, and this result solves the “Bording’s conjecture.”

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