Compact finite difference relaxation method for chaotic and hyperchaotic initial value systems

• Published in: Computational & applied mathematics Vol. 37; no. 4; pp. 5187 - 5202
• Main Authors: Mathale, D., Dlamini, P. G., Khumalo, M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2018; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied physics; Boundary value problems; Chaos theory; Computational mathematics; Computational Mathematics and Numerical Analysis; Finite difference method; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Nonlinearity; Relaxation method (mathematics); Compact finite differences; 65L05; Relaxation; Multi-domain; Chaotic and hyperchaotic systems; 37M05; 65L12
• ISSN: 0101-8205; 2238-3603; 1807-0302
• DOI: 10.1007/s40314-018-0624-4

In this paper, we present a new application of higher order compact finite differences to solve nonlinear initial value problems exhibiting chaotic behaviour. The method involves dividing the domain of the problem into multiple sub-domains, with each sub-domain integrated using higher order compact finite difference schemes. The nonlinearity is dealt using a Gauss–Seidel-like relaxation. The method is, therefore, referred to as the multi-domain compact finite difference relaxation method (MD-CFDRM). In this new application, the MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The results are compared with spectral-based multi-domain method.