86,565 research outputs found
A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods
No full textThe paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method
On conjugate-symplecticity properties of a multi-derivative extension of the midpoint and trapezoidal methods
No full textConjugate symplecticity up to order p 2 of p-th one-step multi-derivative methods based on an extension of the midpoint and trapezoidal methods is proved. If compared with similar achievements obtained for the class of Euler-MacLauren and Hermite-Obreshkov methods, this result further confirm that multi-derivative methods, despite failing in achieving the symplecticity property, may play a significant role in the context of geometric integration. A numerical illustration has also been added
A general framework for solving differential equations
No full textRecently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along a suitable orthonormal basis. Interestingly, this approach can be extended to cope with more general differential problems. In this paper we sketch this fact, by considering some relevant examples
Maximal‐entropy driven determination of weights in least‐square approximation
No full textWe exploit the idea to use the maximal-entropy method, successfully tested in information theory and statistical thermodynamics, to determine approximating function's coefficients and squared errors' weights simultaneously as output of one single problem in least-square approximation. We provide evidence of the method's capabilities and performance through its application to representative test cases by working with polynomials as a first step. We conclude by formulating suggestions for future work to improve the version of the method we present in this paper
Advanced numerical methods in applied sciences
No full textThe use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application
A minimum-time obstacle-avoidance path planning algorithm for unmanned aerial vehicles
No full textIn this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach
A multiregional extension of the SIR model, with application to the COVID-19 spread in Italy
No full textThe paper concerns a new forecast model that includes the class of undiagnosed infected people, and has a multiregion extension, to cope with the in-time and in-space heterogeneity of an epidemic. The model is applied to the SARS-CoV2 (COVID-19) pandemic that, starting from the end of February 2020, began spreading along the Italian peninsula, by first attacking small communities in north regions, and then extending to the center and south of Italy, including the two main islands. It has proved to be a robust and reliable tool for the forecast of the total and active cases, which can be also used to simulate different scenarios. In particular, the model is able to address a number of issues, such as assessing the adoption of the lockdown in Italy, started from March 11, 2020; the estimate of the actual attack rate; and how to employ a rapid screening test campaign for containing the epidemic
A shooting-Newton procedure for solving fractional terminal value problems
No full textIn this paper we consider the numerical solution of fractional terminal value problems: namely, terminal value problems for fractional differential equations. In particular, the proposed method uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems, i.e., initial value problems for fractional differential equations. As a result, the method is able to produce spectrally accurate solutions of fractional terminal value problems. Some numerical tests are reported to make evidence of its effectiveness
(Spectral) Chebyshev collocation methods for solving differential equations
No full textRecently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli (Numer. Algo., 27, 119–130 2021). In this paper, the use of a different framework allows us to carry out a novel analysis of the methods also when they are used as spectral formulae in time, along with some generalizations of the methods
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