An Efficient Spectral Method-based Algorithm for Solving a ‎High-dimensional Chaotic Lorenz System

Document Type : Research Paper


Department of Mathematics and Applied Mathematics, University of Johannesburg, P.O. Box 17011, Doornfontein, 2028, South Africa


In this paper, we implement the multidomain spectral relaxation method to numerically study high dimensional chaos by considering the nine-dimensional Lorenz system. Chaotic systems are characterized by rapidly changing solutions, as well as sensitivity to small changes in initial data. Most of the existing numerical methods converge slowly for this kind of problems and this results in inaccurate approximations. Spectral methods are known for their high accuracy. However, they become less accurate for problems characterised by chaotic solutions, even with an increase in the number of grid points. As a result, in this work, we adopt the multidomain approach which assumes that the main interval can be decomposed into a finite number of subdomains and the solution obtained in each of the subdomains. This approach remarkably improves the results as well as the efficiency of the method.


Main Subjects

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