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Algorithms
Volume 17
Issue 12
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Algorithms, Volume 17, Issue 12 (December 2024) – 2 articles

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24 pages, 1713 KiB  
Article
Stability Optimization of Explicit Runge–Kutta Methods with Higher-Order Derivatives
by Gerasim V. Krivovichev
Algorithms 2024, 17(12), 535; https://doi.org/10.3390/a17120535 (registering DOI) - 21 Nov 2024
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Abstract
The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can [...] Read more.
The paper is devoted to the parametric stability optimization of explicit Runge–Kutta methods with higher-order derivatives. The key feature of these methods is the dependence of the coefficients of their stability polynomials on free parameters. Thus, the integral characteristics of stability domains can be considered as functions of free parameters. The optimization is based on the numerical maximization of the area of the stability domain and the length of the stability interval. Runge–Kutta methods with higher-order derivatives, presented in previous works, are optimized. The optimal values of parameters are computed for methods of fourth, fifth, and sixth orders. In numerical experiments, optimal parameter values are used for the construction of high-order schemes for the method of lines for problems with partial differential equations. Problems for linear and nonlinear hyperbolic and parabolic equations are considered. Additionally, an optimized scheme is used in lattice Boltzmann simulations of gas flow. As the main result of computations and comparison with existing methods, it is demonstrated that optimized schemes have better stability properties and can be used in practice. Full article
(This article belongs to the Section Algorithms for Multidisciplinary Applications)
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11 pages, 5355 KiB  
Article
Adaptive-Step Perturb-and-Observe Algorithm for Multidimensional Phase Noise Stabilization in Fiber-Based Multi-Arm Mach–Zehnder Interferometers
by H. Abarzúa, C. Melo, S. E. Restrepo, S. Vergara, D. Sbarbaro, G. Cañas, G. Lima, G. Saavedra and J. Cariñe
Algorithms 2024, 17(12), 534; https://doi.org/10.3390/a17120534 (registering DOI) - 21 Nov 2024
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Abstract
Fiber-optic Mach–Zehnder interferometers are widely used in research areas such as telecommunications, spectroscopy, and quantum information. These optical structures are known to be affected by phase fluctuations that are usually modeled as multiparametric noise. This multidimensional noise must be stabilized or compensated for [...] Read more.
Fiber-optic Mach–Zehnder interferometers are widely used in research areas such as telecommunications, spectroscopy, and quantum information. These optical structures are known to be affected by phase fluctuations that are usually modeled as multiparametric noise. This multidimensional noise must be stabilized or compensated for to enable fiber-optic Mach–Zehnder architectures for practical applications. In this work, we study the effectiveness of a modified Perturb-and-Observe (P&O) algorithm to control multidimensional phase noise in fiber-based multi-arm Mach–Zehnder interferometers. We demonstrate the feasibility of stabilizing multidimensional phase noise by numerical simulations using a simple feedback control scheme and analyze the algorithm’s performance for systems up to dimension 8×8. We achieved minimal steady-state errors that guarantee high optical visibility in complex optical systems with N×N matrices (with N=[2,3,4,5,6,7,8]). Full article
(This article belongs to the Section Algorithms for Multidisciplinary Applications)
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