On the multi-domain compact finite difference relaxation method for high dimensional chaos: The nine-dimensional Lorenz system

Kouagou, J.N.; Dlamini, Phumlani; Simelane, Simphiwe

doi:10.1016/j.aej.2020.04.025

Cited by 9 publications

(8 citation statements)

References 26 publications

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“…In this section, the results of Taylor's decomposition method (TDM) is presented for the 9D Lorenz system (1) given with the initial condition 𝒙(0) = {0.01,0,0.01,0,0,0,0,0,0.01} and parameters 𝜎 = 0.5, 𝑎 = 0.5, 𝑟 = 14.1 (for chaotic case) and 𝑟 = 55 (for hyperchaotic case). The reason of these choices is to make comparisons with the solutions of the multidomain compact finite difference relaxation method (MD-CFDRM) [3] and Runge-Kutta method (RK4).…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…where 𝒀, 𝒀 0 ∈ ℝ 𝑛 , 𝑛 is the number of differential equations in (2). In [3], it is recommended to use the Taylor's decomposition method in order to approximate (2). Since 9D Lorenz system is in the form (2), in this work the Taylor's decomposition method is suggested to find approximate solution of 9D Lorenz system.…”

Section: Methodsmentioning

confidence: 99%

“…In this work, nine-dimensional (9D) Lorenz system, which is investigated by many authors in recent years, is considered. Some of these works are given in [2][3][4][5][6] Reiterer et al [2] who apply a triple Fourier expansion to the Boussinesq-Oberbeck equations governing thermal convection in a 3D spatial domain by using an approach similar to the well known 3D Lorenz system is given by…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solutions of Nine-Dimensional Lorenz System

Adiyaman¹

2022

Deu Muhendislik Fakultesi Fen Ve Muhendislik

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In this study, the performance of the Taylor's decomposition method in the nine-dimensional Lorenz system is investigated. The proposed method is applied to the nine-dimensional Lorenz system for both chaotic and hyperchaotic cases. Phase portraits and time series graphs of the problem are drawn. The results obtained are compared with both the fourth order Runge-Kutta method and multi-domain compact finite difference relaxation method. Comparisons show that the obtained results are consistent with the results of other methods. Thus, the accuracy and efficiency of the method for nonlinear systems were emphasized by using the nine-dimensional Lorenz system.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical Solutions of Nine-Dimensional Lorenz System

Adiyaman¹

2022

Deu Muhendislik Fakultesi Fen Ve Muhendislik

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“…The main aim of this work is to estimate and give the numerical solution to the COVID-19 model by MSRM using a combination of the Chebyshev spectral method [27][28][29][30][31][32] and the Gauss-Seidel iteration method to address the system in many subdomains that make up the full domain of the model. We can easily predict the feasibility of our numerical method for this model.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique

et al. 2023

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The major objective of this work is to evaluate and study the model of coronavirus illness by providing an efficient numerical solution for this important model. The model under investigation is composed of five differential equations. In this study, the multidomain spectral relaxation method (MSRM) is used to numerically solve the suggested model. The proposed approach is based on the hypothesis that the domain of the problem can be split into a finite number of subintervals, each of which can have a solution. The procedure also converts the proposed model into a system of algebraic equations. Some theoretical studies are provided to discuss the convergence analysis of the suggested scheme and deduce an upper bound of the error. A numerical simulation is used to evaluate the approach’s accuracy and utility, and it is presented in symmetric forms.

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“…Dlamini et al 29 used them to solve ODEs in fluid mechanics. They have also been used to solve chaotic initial value problems by Mathale et al 30 and Kouagou et al 31 Zhao and Corless used the CFDSs to solve integro-differential equations 32 .…”

Section: Introductionmentioning

confidence: 99%

Solving singular boundary value problems using higher-order compact finite difference schemes with a novel higher-order implementation of Robin boundary conditions

Dlamini

Malele

Simelane

2022

Preprint

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This study introduces a novel and very accurate method for implementing Robin boundary conditions while solving boundary value problems with compact finite difference schemes. Most studies numerically approximate Neumann and Robin boundary conditions with a first-order approximation, however, this reduces the accuracy in general. Our implementation produces highly accurate results in solving nonlinear singular boundary value problems. Other reported results achieved utilizing various techniques are used for comparison. Tables and graphs showing the results are given.

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On the multi-domain compact finite difference relaxation method for high dimensional chaos: The nine-dimensional Lorenz system

Cited by 9 publications

References 26 publications

Numerical Solutions of Nine-Dimensional Lorenz System

Numerical Solutions of Nine-Dimensional Lorenz System

Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique

Solving singular boundary value problems using higher-order compact finite difference schemes with a novel higher-order implementation of Robin boundary conditions

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