scholarly journals Solving Advection Equations by Applying the Crank-Nicolson Scheme Combined with the Richardson Extrapolation

2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Zahari Zlatev ◽  
Ivan Dimov ◽  
István Faragó ◽  
Krassimir Georgiev ◽  
Ágnes Havasi ◽  
...  
Keyword(s):  
Mathematical Models ◽  
Numerical Solution ◽  
Large Scale ◽  
Richardson Extrapolation ◽  
Numerical Treatment ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Nicolson Scheme ◽  
Crank Nicolson

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations will be formulated and proved in this paper. The usefulness of the combination consisting of the Crank-Nicolson scheme and the Richardson Extrapolation will be illustrated by numerical examples.

Beam Propagation Method Based on the Iterated Crank–Nicolson Scheme for Solving Large-Scale Wave Propagation Problems

2015 ◽  
Vol 51 (3) ◽  
pp. 1-4 ◽  
Author(s):  
Dimitra A. Ketzaki ◽  
Ioannis T. Rekanos ◽  
Theodoros I. Kosmanis ◽  
Traianos V. Yioultsis
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Beam Propagation Method ◽  
Beam Propagation ◽  
Nicolson Scheme ◽  
Crank Nicolson

scholarly journals Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation

2019 ◽  
Vol 9 (5) ◽  
pp. 3720
Author(s):  
Kamel Al-Khaled ◽  
Issam Abu-Irwaq
Keyword(s):  
Analytical Solution ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Present Article ◽  
Point Theory ◽  
Algebraic Equations ◽  
Time Direction ◽  
Numerical Examples ◽  
Sivashinsky Equation ◽  
Crank Nicolson

The present article is designed to supply two different numerical<br />solutions for solving  Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for  Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

scholarly journals A B-Spline Quasi Interpolation Crank–Nicolson Scheme for Solving the Coupled Burgers Equations with the Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
M. Taghipour ◽  
H. Aminikhah
Keyword(s):  
Algebraic Equations ◽  
Unconditionally Stable ◽  
B Spline ◽  
Numerical Examples ◽  
Burgers Equations ◽  
Second Derivatives ◽  
Spatial Derivatives ◽  
Nicolson Scheme ◽  
Derivatives Of ◽  
Crank Nicolson

In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.

scholarly journals Numerical solution of advection-diffusion equation using finite difference schemes

2020 ◽  
Vol 55 (1) ◽  
pp. 15-22
Author(s):  
LS Andallah ◽  
MR Khatun
Keyword(s):  
Diffusion Equation ◽  
Numerical Solution ◽  
Numerical Scheme ◽  
Finite Difference Schemes ◽  
Qualitative Behavior ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nicolson Scheme ◽  
And Diffusion ◽  
Crank Nicolson

This paper presents numerical simulation of one-dimensional advection-diffusion equation. We study the analytical solution of advection diffusion equation as an initial value problem in infinite space and realize the qualitative behavior of the solution in terms of advection and diffusion co-efficient. We obtain the numerical solution of this equation by using explicit centered difference scheme and Crank-Nicolson scheme for prescribed initial and boundary data. We implement the numerical scheme by developing a computer programming code and present the stability analysis of Crank-Nicolson scheme for ADE. For the validity test, we perform error estimation of the numerical scheme and presented the numerical features of rate of convergence graphically. The qualitative behavior of the ADE for different choice of the advection and diffusion co-efficient is verified. Finally, we estimate the pollutant in a river at different times and different points by using these numerical scheme. Bangladesh J. Sci. Ind. Res.55(1), 15-22, 2020

scholarly journals Study on Accuracy of the High-Resolution Schemes

2014 ◽  
Vol 6 ◽  
pp. 905053
Author(s):  
Yawen Tang ◽  
Bo Yu ◽  
Jianyu Xie ◽  
Jingfa Li ◽  
Peng Wang
Keyword(s):  
High Resolution ◽  
Numerical Solution ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
High Resolution Schemes ◽  
Artificial Diffusion ◽  
Convection Dominated

The high-resolution (HR) schemes have been widely used as they can achieve the numerical solution without oscillation and artificial diffusion, especially for convection-dominated problems. However, there still have arguments about the order of accuracy of HR schemes, especially about the extreme value of the solution. In this paper, it is proved that any HR scheme designed in the NVD diagram has second-order accuracy when its combined segments totally locate in the BAIR region. In other words, it has been verified in our study that the segments, which have low-order accuracy when independently employed, have at least second-order accuracy when locate in BAIR region by analysis of two implementation methods of HR scheme and also a number of numerical examples. Meanwhile Richardson extrapolation has been used to estimate the order of accuracy of HR schemes which achieve the same conclusion.

scholarly journals Efficient implementation of advanced Richardson Extrapolation in an atmospheric chemical scheme

2021 ◽  
Author(s):  
Zahari Zlatev ◽  
Ivan Dimov ◽  
István Faragó ◽  
Krassimir Georgiev ◽  
Ágnes Havasi
Keyword(s):  
Large Scale ◽  
Chemical Species ◽  
Richardson Extrapolation ◽  
Computational Time ◽  
Numerical Treatment ◽  
Variable Stepsize ◽  
Pollution Levels ◽  
Environmental Models ◽  
Diffusion Phenomena ◽  
Chemical Scheme

AbstractThe numerical treatment of an atmospheric chemical scheme, which contains 56 species, is discussed in this paper. This scheme is often used in studies of air pollution levels in different domains, as, for example, in Europe, by large-scale environmental models containing additionally two other important physical processes—transport of pollutants in the atmosphere (advection) and diffusion phenomena. We shall concentrate our attention on the efficient numerical treatment of the chemical scheme by using Implicit Runge–Kutta Methods combined with accurate and efficient advanced versions of the Richardson Extrapolation. A Variable Stepsize Variable Formula Method is developed in order to achieve high accuracy of the calculated results within a reasonable computational time. Reliable estimations of the computational errors when the proposed numerical methods are used in the treatment of the chemical scheme will be demonstrated by presenting results from several representative runs and comparing these results with “exact” concentrations obtained by applying a very small stepsize during the computations. Results related to the diurnal variations of some of the chemical species will also be presented. The approach used in this paper does not depend on the particular chemical scheme and can easily be applied when other atmospheric chemical schemes are selected.

Using the generalized Adams-Bashforth-Moulton method for obtaining the numerical solution of some variable-order fractional dynamical models

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed M. Khader
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Numerical Results ◽  
Convergence Order ◽  
Variable Order ◽  
Numerical Treatment ◽  
Dynamical Models ◽  
Fractional Modeling ◽  
Dynamics Problems

AbstractThis paper is devoted to introduce a numerical treatment using the generalized Adams-Bashforth-Moulton method for some of the variable-order fractional modeling dynamics problems, such as Riccati and Logistic differential equations. The fractional derivative is described in Caputo variable-order fractional sense. The obtained numerical results of the proposed models show the simplicity and efficiency of the proposed method. Moreover, the convergence order of the method is also estimated numerically.

Perturbation Bounds and Condition Numbers for a Complex Indefinite Linear Algebraic System

2016 ◽  
Vol 6 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Lei Zhu ◽  
Wei-Wei Xu ◽  
Xing-Dong Yang
Keyword(s):  
Algebraic System ◽  
Condition Numbers ◽  
Linear Algebraic System ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Interest In Science ◽  
Perturbation Bounds

AbstractWe consider perturbation bounds and condition numbers for a complex indefinite linear algebraic system, which is of interest in science and engineering. Some existing results are improved, and illustrative numerical examples are provided.

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