scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

scholarly journals Time-step limits for stable solutions of the ice-sheet equation

1996 ◽  
Vol 23 ◽  
pp. 74-85 ◽  
Author(s):  
Richard C. A. Hindmarsh ◽  
Antony J. Payne
Keyword(s):  
Linear Equations ◽  
Computational Cost ◽  
Ice Sheet ◽  
Two Dimensions ◽  
Picard Iteration ◽  
Time Step ◽  
Iterated Maps ◽  
A New Technique ◽  
The Stability ◽  
Non Linear Equations

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

Dynamic modelling of planar flexible manipulators: Computational and algorithmic efficiency

1997 ◽  
Vol 211 (2) ◽  
pp. 119-133 ◽  
Author(s):  
Y-J Shyu ◽  
K F Gill
Keyword(s):  
Linear Equations ◽  
Dynamic Modelling ◽  
Approximate Solutions ◽  
Experimental Information ◽  
Open Loop ◽  
Flexible Manipulator ◽  
Time Step ◽  
End Effector ◽  
And Control ◽  
Non Linear Equations

Traditionally, many robot arms are very rigid in construction; this was believed to be necessary for accurate placement and repeatability but led to higher material costs and increased energy consumption. Higher operational speeds and the use of lightweight materials cause elastic deformations to occur during the operation of the manipulator. These deformations degrade the path-tracking performance of the end-effector. The dynamic behaviour of a flexible manipulator is described mathematically by non-linear equations which are difficult to solve analytically. Unfortunately, there is currently no experimental information available with which to compare this design of flexible structure. For design and control purposes, it is suggested in this paper that it is more appropriate to employ approximate solutions with the emphasis on the development of a fast computational algorithm. An analytical study was undertaken to investigate the relevant uncertainties that are either inappropriately described or unavailable in the literature. The purpose of the paper is essentially to include the initial deflections in the simulation, to select the size of the time step, to select the models for emulating the end-effector, payload and joint actuator and, finally, to suppress the uncontrollable off-plane vibrations when encountered. When this knowledge has been obtained, the design and development of the simulation process can begin. In order to demonstrate the practicability of the open-loop simulation proposed and test the software, two representative models were investigated.

scholarly journals On Efficient Iterative Numerical Methods for Simultaneous Determination of all Roots of Non-Linear Function

2020 ◽  
Author(s):  
Mudassir Shams ◽  
Nazir Mir ◽  
Naila Rafiq
Keyword(s):  
Simultaneous Determination ◽  
Iterative Methods ◽  
Linear Equations ◽  
Basin Of Attraction ◽  
Numerical Test ◽  
Computational Cost ◽  
Single Root ◽  
Non Linear ◽  
Non Linear Equations

We construct a family of two-step optimal fourth order iterative methods for finding single root of non-linear equations. We generalize these methods to simultaneous iterative methods for determining all the distinct as well as multiple roots of single variable non-linear equations. Convergence analysis is present for both cases to show that the order of convergence is four in case of single root finding method and is twelve for simultaneous determination of all roots of non-linear equation. The computational cost, Basin of attraction, efficiency, log of residual and numerical test examples shows, the newly constructed methods are more efficient as compared to the existing methods in literature.

Stability and vibration of a non-linear vehicle and passenger system

2002 ◽  
Vol 216 (2) ◽  
pp. 109-116 ◽  
Author(s):  
K Yu ◽  
A C J Luo ◽  
Y He
Keyword(s):  
Equations Of Motion ◽  
Linear Equations ◽  
Dynamic Responses ◽  
Spring Stiffness ◽  
Torsional Spring ◽  
Non Linear ◽  
Linear Equations Of Motion ◽  
Safety Belts ◽  
The Stability ◽  
Non Linear Equations

A non-linear dynamic model to predict the passenger's response in a vehicle travelling on a rough pavement surface (or a rough terrain) is developed. The corresponding equilibrium and stability are investigated through the non-linear equations of motion for a vehicle and passenger system with impacts. The stability with respect to the torsional spring stiffness of safety belts is illustrated. Based on such a stability condition, the dynamic responses for the vehicle and passenger system with and without impacts are simulated numerically. This investigation shows that a strong torsional spring is required in order to reduce the vibration amplitudes of passengers and to avoid impacts between the vehicle and passenger.

scholarly journals Linear computational cost implicit solver for elliptic problems

2020 ◽  
Vol 21 (3) ◽  
Author(s):  
Grzegorz Gurgul ◽  
Marcin Los ◽  
Maciej Paszynski ◽  
Victor Calo
Keyword(s):  
Heat Transfer ◽  
Transfer Problem ◽  
Linear Equations ◽  
Computational Cost ◽  
Basis Functions ◽  
Implicit Method ◽  
Time Step ◽  
B Spline ◽  
Spline Basis ◽  
Intermediate Time

In this paper, we use the alternating direction method for isogeometric finite elements to simulate implicit dynamics. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit marching method to fully discretize the problem. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along a particular coordinate axis in the intermediate time steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of this algebraic transformations, we get a system of linear equations that can be factorized in linear $O(N)$ computational cost in every time step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

scholarly journals Numerical solutions of the generalized Burgers-Huxley equation by implicit exponential finite difference method

2015 ◽  
Vol 11 (2) ◽  
pp. 57-67 ◽  
Author(s):  
B. İnan ◽  
A. R. Bahadir
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Linear Equations ◽  
System Of Nonlinear Equations ◽  
Time Step ◽  
Gauss Elimination ◽  
Huxley Equation ◽  
A New Technique

Abstract In this paper, numerical solutions of the generalized Burgers-Huxley equation are obtained using a new technique of forming improved exponential finite difference method. The technique is called implicit exponential finite difference method for the solution of the equation. Firstly, the implicit exponential finite difference method is applied to the generalized Burgers-Huxley equation. Since the generalized Burgers-Huxley equation is nonlinear the scheme leads to a system of nonlinear equations. Secondly, at each time-step Newton’s method is used to solve this nonlinear system then linear equations system is obtained. Finally, linear equations system is solved using Gauss elimination method at each time-step. The numerical solutions obtained by this way are compared with the exact solutions and obtained by other methods to show the efficiency of the method.

scholarly journals One-dimensional pattern formation in a model of burning

1993 ◽  
Vol 35 (2) ◽  
pp. 145-173 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Thermal Conduction ◽  
Linear Equations ◽  
Series Representation ◽  
Reaction Diffusion ◽  
Temperature Sensitive ◽  
Governing Equations ◽  
Second Stage ◽  
Nonuniqueness Of Solutions ◽  
The Stability ◽  
Non Linear Equations

AbstractWe discuss a model of a burning process, essentially due to Sal'nikov, in which a substrate undergoes a two-stage decay through some intermediate chemical to form a final product. The second stage of the process occurs at a temperature-sensitive rate, and is also responsible for the production of heat. The effects of thermal conduction are included, and the intermediate chemical is assumed to be capable of diffusion through the decomposing substrate. The governing equations thus form a reaction-diffusion system, and spatially inhomogeneous behaviour is therefore possible.This paper is concerned with stationary patterns of temperature and chemical concentration in the model. A numerical method for the solution of the governing equations is outlined, and makes use of a Fourier-series representation of the pattern. The question of the stability of these patterns is discussed in detail, and a linearised solution is presented, which is valid for patterns of very small amplitude. The results of accurate solutions to the fully non-linear equations are discussed, and compared with the predictions of the linearised theory. Parameter regions in which there exists genuine nonuniqueness of solutions are identified.

scholarly journals Error analysis of nonlinear time fractional mobile/immobile advection-diffusion equation with weakly singular solutions

2021 ◽  
Vol 24 (1) ◽  
pp. 202-224
Author(s):  
Hui Zhang ◽  
Xiaoyun Jiang ◽  
Fawang Liu
Keyword(s):  
Computational Cost ◽  
Correction Method ◽  
Trapezoidal Rule ◽  
Stability And Convergence ◽  
Fast Method ◽  
Memory Requirement ◽  
Time Step ◽  
Advection Dispersion ◽  
Advection Diffusion Equation ◽  
The Stability

Abstract In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. The memory requirement and computational cost are O(Q) and O(QK), respectively, where K is the number of the final time step and Q is the number of quadrature points used in the trapezoidal rule. Some numerical experiments are given to confirm our theoretical analysis and the effectiveness of the presented methods.

k-t scattering formulation of the absorptive acoustic wave equation: Wraparound and edge‐effect elimination

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2185-2192 ◽  
Author(s):  
B. Compani‐Tabrizi
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Two Dimensions ◽  
Time Step ◽  
Absorbing Boundary ◽  
Temporal Derivative ◽  
Time Domain Scattering ◽  
The Stability ◽  
Spatially Varying ◽  
And Storage

The solution algorithm to the absorptive acoustic scalar wave equation with spatially varying velocity and absorptive fields is numerically examined in the context of the k-space time‐domain scattering formalism to construct an absorbing boundary potential which eliminates wraparound and edge effects. The absorptive potential is constructed by using the absorptive coefficient, i.e., the coefficient of the first temporal derivative in the differential equation. Numerical solutions, in two dimensions, show the stability of the algorithm and the elimination of wraparound and edge reflections through use of the constructed absorptive potential. The numbers of calculations and storage requirements per time step are on the order of [Formula: see text] and N, respectively, where N is the number of points into which the problem is discretized.

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