scholarly journals Study on Truncation and Roundoff Errors in Numerical Methods

2021 ◽  
Vol 23 (08) ◽  
pp. 143-147
Author(s):  
Keyword(s):  
Numerical Calculation ◽  
Numerical Methods ◽  
Truncation Error ◽  
Stock Exchange ◽  
Parliamentary Elections ◽  
Roundoff Errors ◽  
Short Flight

History is full of many examples where errors in numerical calculation have played important role or sometimes proved fatal also. (e.g. The Patriot and the Scud [6], The short flight of the Ariane 5 [7], The Vancouver Stock Exchange [8,9,10], Parliamentary elections in Schleswig-Holstein [11]). So in this paper we review two types of errors occurred while performing calculations: first is truncation error and second is round off error . Examples are given in support of the theory part.

scholarly journals An energy concerving modification of numerical methods for the integration of equations of motion

1987 ◽  
Vol 10 (1) ◽  
pp. 173-179
Author(s):  
Robert A. Labudde ◽  
Donald Greenspan
Keyword(s):  
Numerical Methods ◽  
Total Energy ◽  
Truncation Error ◽  
Equations Of Motion ◽  
Simple Modification ◽  
System Of Particles

In the integration of the equations of motion of a system of particles, conventional numerical methods generate an error in the total energy of the same order as the truncation error. A simple modification of these methods is described, which results in exact conservation of the energy.

scholarly journals Numerical Methods for Advection Problem

2020 ◽  
Vol 13 (1) ◽  
pp. 144-157
Author(s):  
Diogene Vianney Pongui Ngoma ◽  
Germain Nguimbi ◽  
Vital Delmas Mabonzo ◽  
Narcisse Batangouna
Keyword(s):  
Mathematical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Truncation Error ◽  
Numerical Techniques ◽  
Analytical Stability

This work is part of mathematical modeling and numerical analysis. This paper aims is to solve an advection problem where u=u(x; t) is the solution by Lax-Wendrof and nite dierence methods, to study the analytical stability in L2[0;1], L1[0; 1], then calculate the truncation error of these methods and nally study the analytical convergence of these methods. These numerical techniques of resolution were implemented in Scilab.

Comparison of two numerical methods for the integration of the Takagi–Taupin equations

1981 ◽  
Vol 14 (6) ◽  
pp. 432-436 ◽  
Author(s):  
C. Nourtier ◽  
D. Taupin
Keyword(s):  
Numerical Calculation ◽  
Numerical Methods ◽  
Two Dimensions ◽  
Integration Step ◽  
Order Method ◽  
Third Order ◽  
Acta Cryst ◽  
One Dimensional ◽  
Numerical Resolution ◽  
The One

Two methods for the numerical resolution of the Takagi-Taupin equations are compared. It is shown that for a small integration step Taupin's [Acta Cryst. (1967), 23, 25–35] extension to two dimensions of the one-dimensional Runge–Kutta third-order method is more accurate than the algorithm of Authier, Malgrange & Tournarie [Acta Cryst. (1968), A24, 126–136] but, for a given precision, Authier, Malgrange & Tournarie's method is faster than Taupin's so the former will usually be preferred for numerical calculation.

scholarly journals Numerical methods for the multiplicative partial differential equations

2017 ◽  
Vol 15 (1) ◽  
pp. 1344-1350
Author(s):  
Muhammet Yazıcı ◽  
Harun Selvitopi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Methods ◽  
Error Estimation ◽  
Truncation Error ◽  
Numerical Solutions ◽  
Partial Differential ◽  
The Matrix ◽  
The Stability ◽  
Truncation Error Estimation

Abstract We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

scholarly journals Calculation of the foundation settlement during compensation grouting

2020 ◽  
Vol 8 (1) ◽  
pp. 33-38
Author(s):  
Ivan Luzin
Keyword(s):  
Numerical Calculation ◽  
Hydrostatic Pressure ◽  
Numerical Methods ◽  
Foundation Settlement ◽  
Volume Strain ◽  
3D Design ◽  
Plaxis 3D

The article presents the issue of correctly calculating the SSC of the foundation bases in the process of compensation grouting. It is shown that the numerical calculation should be performed in two versions: hydrostatic pressure and rear volume strain in a cluster of soil. Using the example of an object in Moscow, the use of the PLAXIS 3D design complex was shown to calculate additional precipitation of buildings during compensation grouting and the passage of a tunnel under it. A method is given for determining the required volume of suspension during compensation grouting using numerical methods for calculating the SSC.

A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows

2014 ◽  
Vol 24 (3) ◽  
pp. 735-759 ◽  
Author(s):  
Anis Younes ◽  
Ahmed Makradi ◽  
Ali Zidane ◽  
Qian Shao ◽  
Lyazid Bouhala
Keyword(s):  
Natural Convection ◽  
Numerical Methods ◽  
Analytical Solution ◽  
Discontinuous Galerkin ◽  
Truncation Error ◽  
Computational Time ◽  
Time Step ◽  
Content Type ◽  
Flow Problems ◽  
Time Stepping

Purpose – The purpose of this paper is to develop an efficient non-iterative model combining advanced numerical methods for solving buoyancy-driven flow problems. Design/methodology/approach – The solution strategy is based on two independent numerical procedures. The Navier-Stokes equation is solved using the non-conforming Crouzeix-Raviart (CR) finite element method with an upstream approach for the non-linear convective term. The advection-diffusion heat equation is solved using a combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods. To reduce the computational time due to the coupling, the authors use a non-iterative time stepping scheme where the time step length is controlled by the temporal truncation error. Findings – Advanced numerical methods have been successfully combined to solve buoyancy-driven flow problems on unstructured triangular meshes. The accuracy of the results has been verified using three test problems: first, a synthetic problem for which the authors developed a semi-analytical solution; second, natural convection of air in a square cavity with different Rayleigh numbers (103-108); and third, a transient natural convection problem of low Prandtl fluid with horizontal temperature gradient in a rectangular cavity. Originality/value – The proposed model is the first to combine advanced numerical methods (CR, DG, MPFA) for buoyancy-driven flow problems. It is also the first to use a non-iterative time stepping scheme based on local truncation error control for such coupled problems. The developed semi analytical solution based on Fourier series is also novel.

Rapidly Analyze the Fault Tree with Numerical Methods

2014 ◽  
Vol 596 ◽  
pp. 209-215
Author(s):  
Bing Wu ◽  
Zi Hao Zhao ◽  
Yu Hui Ren
Keyword(s):  
Numerical Calculation ◽  
Numerical Methods ◽  
Qualitative Analysis ◽  
Prime Number ◽  
Fault Tree ◽  
Fault Tree Analysis ◽  
New Method ◽  
Minimum Cut ◽  
Tree Analysis

In order to take advantage of the computer in the numerical calculation, in view of the fault tree qualitative analysis, a new method of fault tree analysis, numerical methods, was proposed on the basis of absorbing merits of the prime number method and determinant method. The fault tree's minimum cut sets and path sets can be calculated effectively with this method. Consequently, a specific example about calculation of the minimum cut sets was demonstrated with Scilab in this paper.

scholarly journals Numerical Methods for Advection Problem

2020 ◽  
Vol 13 (1) ◽  
pp. 144-157
Author(s):  
Diogene Vianney Pongui Ngoma ◽  
Germain Nguimbi ◽  
Vital Delmas Mabonzo ◽  
Narcisse Batangouna
Keyword(s):  
Mathematical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Truncation Error ◽  
Numerical Techniques ◽  
Analytical Stability

This work is part of mathematical modeling and numerical analysis. This paper aims is to solve an advection problem where u=u(x; t) is the solution by Lax-Wendrof and nite dierence methods, to study the analytical stability in L2[0;1], L1[0; 1], then calculate the truncation error of these methods and nally study the analytical convergence of these methods. These numerical techniques of resolution were implemented in Scilab.

DETERMINING THE COMPOSITION OF A PORTFOLIO MANAGEMENT FROM A GROUPINGS MODEL

2013 ◽  
Vol 21 (04) ◽  
pp. 561-578 ◽  
Author(s):  
ANNA M. GIL-LAFUENTE ◽  
JAIME GIL-ALUJA ◽  
LUCIANO BARCELLOS DE PAULA
Keyword(s):  
Numerical Methods ◽  
Portfolio Management ◽  
Stock Exchange ◽  
Non Linear ◽  
Dual Perspective

Often, in situations of uncertainty in portfolio management, it is difficult to apply the numerical methods based on the linearity principle. When this happens it is possible to use nonnumeric techniques to assess the situations with a non linear attitude. One of the concepts that can be used in these situations is the concept of grouping. In the last thirty years, several studies have tried to give good solutions to the problems of homogeneous groupings. For example, we could mention the Pichat algorithm, the affinities algorithms and several studies developed by the authors of this work. In this paper, we use some topological axioms in order to develop an algorithm that is able to reduce the number of elements of the power sets of the related sets by connecting them to the sets that form the topologies. We will apply this algorithm in the grouping of titles listed in the Stock Exchange or in its dual perspective.

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