Non-standard finite-difference methods for vibro-impact problems

2005 ◽  
Vol 461 (2058) ◽  
pp. 1927-1950 ◽  
Author(s):  
Yves Dumont ◽  
Jean M.-S Lubuma
Keyword(s):  
Finite Difference ◽  
Duffing Oscillator ◽  
Finite Difference Methods ◽  
Finite Difference Schemes ◽  
Restitution Coefficient ◽  
Conservation Of Energy ◽  
Numerical Examples ◽  
Impact Oscillators ◽  
Difference Methods ◽  
Impact Problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

scholarly journals Finite Difference Methods for Weather Prediction in Abuja, Nigeria

2020 ◽  
pp. 915-931
Author(s):  
Jacob Emmanuel ◽  
Ogunfiditimi F.O. ◽  
Victor Alexander Okhuese ◽  
Odeyemi J. K
Keyword(s):  
Finite Difference ◽  
Weather Prediction ◽  
Finite Difference Methods ◽  
Weather Forecast ◽  
Difference Schemes ◽  
Weather Data ◽  
Finite Difference Schemes ◽  
Numerical Weather ◽  
Difference Methods ◽  
The Finite Difference Method

In this research, we have been able to simulate some finite difference schemes to predict weather trends of Abuja Station, Nigeria. By analyzing the results from these schemes, it has shown that the best scheme in the finite difference method that gives a close accurate weather forecast is the trapezoidal scheme hence we use it to simulate numerical weather data obtained from Federal Airports Authority of Nigeria (FAAN), Abuja and corresponding numerical weather data obtained by the compatible finite difference schemes, using MATLAB (R2012a) software to obtain future numerical weather trends.

NUMERICAL INTERFACES IN FINITE DIFFERENCE METHODS FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

1993 ◽  
Vol 01 (02) ◽  
pp. 151-184 ◽  
Author(s):  
TAO LIN
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Discontinuous Coefficients ◽  
Interface Problems ◽  
Artificial Boundary ◽  
Numerical Examples ◽  
Difference Methods

In this paper, we discuss the interface problems arising in using finite difference methods to solve hyperbolic equations with discontinuous coefficients. The schemes developed here can be used to handle four important types of numerical interfaces due to: (1) the discontinuity of the coefficients of the PDE, (2) using artificial boundary, (3) using different finite difference formulae in different areas, and (4) using different grid sizes in different areas. Stability analysis for these schemes is carried out in terms of conventional l1, l2, and l∞ norms so that the convergence rates of these schemes are obtained. Several numerical examples are supplied to demonstrate properties of these schemes.

Analysis of Elastic-Plastic Impact Involving Severe Distortions

1976 ◽  
Vol 43 (3) ◽  
pp. 439-444 ◽  
Author(s):  
G. R. Johnson
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Triangular Element ◽  
Element Formulation ◽  
Lagrangian Analysis ◽  
Triangular Finite Element ◽  
Elastic Plastic ◽  
Analysis Technique ◽  
Difference Methods ◽  
Impact Problems

A Lagrangian analysis technique is presented for two-dimensional axi-symmetric impact problems involving elastic-plastic flow. This technique is based on a triangular finite-element formulation rather than the quadrilateral formulation generally used in comparable finite-difference methods. For impact problems involving severe distortions, the triangular element formulation is better suited to represent the severe distortions than is the traditional quadrilateral finite-difference method. Included are the formulation of the technique and illustrative examples.

scholarly journals Simple bespoke preservation of two conservation laws

2018 ◽  
Vol 40 (2) ◽  
pp. 1294-1329 ◽  
Author(s):  
Gianluca Frasca-Caccia ◽  
Peter Ellsworth Hydon
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Vries Equation ◽  
Finite Difference Methods ◽  
Nonlinear Heat Equation ◽  
Finite Difference Schemes ◽  
Simplified Approach ◽  
Numerical Tests ◽  
Difference Methods ◽  
Discrete Conservation Laws

Abstract Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic–numeric approach feasible. To illustrate the simplified approach we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg–de Vries equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared with others in the literature.

scholarly journals Efficient algorithms for solving nonlinear Volterra integro-differential equations with two point boundary conditions

1998 ◽  
Vol 21 (4) ◽  
pp. 755-760 ◽  
Author(s):  
L. E. Garey ◽  
R. E. Shaw
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Iterative Procedure ◽  
Algebraic System ◽  
Finite Difference Methods ◽  
Efficient Algorithms ◽  
Point Boundary ◽  
Nonlinear Case ◽  
Numerical Examples ◽  
Difference Methods

In this paper a collection of efficient algorithms are described for solving an algebraic system with a symmetric Toeplitz coecient matrix. Systems of this form arise when approximating the solution of boundary value Volterra integro-differential equations with finite difference methods. In the nonlinear case, an iterative procedure is required and is incorporated into the algorithms presented. Numerical examples illustrate the results.

scholarly journals Generating Solar Sail Trajectories in the Earth-Moon System Using Augmented Finite-Difference Methods

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Geoffrey G. Wawrzyniak ◽  
Kathleen C. Howell
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Orbital Plane ◽  
Solar Sail ◽  
Finite Difference Schemes ◽  
Trajectory Design ◽  
Dynamical Regime ◽  
The Earth ◽  
Collocation Scheme ◽  
Difference Methods

Using a solar sail, a spacecraft orbit can be offset from a central body such that the orbital plane is displaced from the gravitational center. Such a trajectory might be desirable for a single-spacecraft relay to support communications with an outpost at the lunar south pole. Although trajectory design within the context of the Earth-Moon restricted problem is advantageous for this problem, it is difficult to envision the design space for offset orbits. Numerical techniques to solve boundary value problems can be employed to understand this challenging dynamical regime. Numerical finite-difference schemes are simple to understand and implement. Two augmented finite-difference methods (FDMs) are developed and compared to a Hermite-Simpson collocation scheme. With 101 evenly spaced nodes, solutions from the FDM are locally accurate to within 1740 km. Other methods, such as collocation, offer more accurate solutions, but these gains are mitigated when solutions resulting from simple models are migrated to higher-fidelity models. The primary purpose of using a simple, lower-fidelity, augmented finite-difference method is to quickly and easily generate accurate trajectories.

scholarly journals Preconditioning in a wavelet basis and its application to some boundary value problems

2003 ◽  
pp. 86-93
Author(s):  
Marija Rasajski
Keyword(s):  
Finite Difference ◽  
Condition Number ◽  
Iterative Procedure ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Condition Numbers ◽  
Wavelet Basis ◽  
Numerical Examples ◽  
The Matrix ◽  
Difference Methods

Standard finite difference methods applied to the boundary value problem a(x)u" (x) + b(x)u'(x) + c(x)u(x) = f (x), u(0) = 0, u(1) = 0, lead to linear systems with large condition numbers. Solving a system, i.e. finding the inverse of a matrix with a large condition number can be achieved by some iterative procedure in a large number of iteration steps. By projecting the matrix of the system into the wavelet basis, and applying a diagonal pre-conditioner, we obtain a matrix with a small condition number. Computing the inverse of such a matrix requires fewer iteration steps, and that number does not grow significantly with the size of the system. Numerical examples, with various operators, are presented to illustrate the effect preconditioners have on the condition number, and the number of iteration steps.

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