1,721,036 research outputs found
Conservazione numerica di proprietà qualitative delle soluzioni di problemi differenziali ed integrali
No full textDiscretized multistep collocation methods for Volterra integro-differential equations
No full textVolterra Integro-Differential Equations (VIDEs) have been proposed as the mathematical models of a wide class of evolutionary problems with memory, such as viscoelastic materials with memory, in which the stress at each point depends both on the present value of the deformation gradient and on the entire temporal history of motion. In this talk we present a promising numerical approach based on multistep collocation. Recently have been introduced exact multistep collocation methods which approximate the solution by a piecewise algebraic polynomial, which satisfies the VIDE on the collocation points. The collocation polynomial depends on a fixed number of previous time-steps (instead of only one, as in the classical one-step methods). These methods have higher order of convergence at the same computational cost of one-step collocation ones, have strong stability properties, and provide an approximation of the solution at each point of the time interval, that is quite useful in a variable step implementation. On the other side, they cannot be directly implemented since they require the evaluation of some integrals. Here we describe the construction of discretized multistep collocation methods, where the collocation polynomial is derived by using suitable quadrature formulas for the approximation of such integrals. We analyze the convergence and the numerical stability of the proposed methods. In particular we establish how to choose quadrature rules which preserve the order of convergence of exact methods, find classes of A0-stable methods and derive an estimate of the local error. We show the performances of our methods on some significative examples
Stability analysis of spline collocation methods for fractional differential equations
Full text linkThis paper deals with spline collocation methods for fractional differential equations, introduced by Pedas and Tamme (2014). Some practical formulas are derived, for the computation of fractional integrals involved in the method, useful for implementation. Linear stability analysis is carried out and stability regions of several methods are provided. Numerical experiments on linear and nonlinear test problems confirm theoretical expectations
Laboratorio di Matematica Computazionale: fare i conti con il computer evitando disastri
Full text linkIl presente lavoro illustra una parte delle attività svolte durante il Laboratorio di Matematica Computazionale, nell’ambito del Progetto Lauree Scientifiche, presso il Dipartimento di Matematica dell’Università di Salerno, rivolto a studenti del quarto e quinto anno delle scuole superiori. L’obiettivo del Laboratorio di Matematica Computazionale è descrivere la risoluzione di un problema reale mediante l’utilizzo del calcolatore. Attraverso esempi semplici si illustra la costruzione di un modello matematico, il passaggio da questo al modello numerico ed infine all’algoritmo ed al software matematico. In particolare, qui viene trattata la risoluzione di sistemi lineari mediante il metodo di eliminazione di Gauss e le relative strategie attuabili al fine di evitare disastri computazionali
Nordsieck methods for non-stiff ordinary differential equations
No full textGeneral linear methods are a wide class of numerical methods for solving ordinary differential systems which includes many classical methods, such as for example Runge-Kutta or linear multistep methods. We describe the construction of explicit methods in Nordsieck form with the so-called quadratic stability (QS), i.e. methods with stability function of the form
p(w, z) = w^(r-1) (w^2 - p1(z)w + p0(z));
where p1(z) and p0(z) are polynomials of z. After satisfying order conditions and criteria that guarantee quadratic form of its stability function, the remaining free parameters of the methods are used to maximize region of absolute stability. We discuss the error estimation formulas and step changing strategies for constructed methods. The results of numerical experiments with variable stepsize code are also presented.
REFERENCES
[1] M. Bras. Nordsieck Methods with Inherent Quadratic Stability. Math. Model. Anal., 16 (1): 82-96.
[2] M. Bras and A. Cardone. Construction of Efficient General Linear Methods for Non-Stiff Differential Systems. Math. Model. Anal., 17 (2): 171-189.
[3] Z. Bartoszewski and Z. Jackiewicz. Explicit Nordsieck methods with extended stability regions. Appl. Math. Comput., 218 6056-6066.
[4] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New
Jersey, 2009
Continuous numerical methods for Volterra Integro-Differential Equations
No full textThe aim of our research is the construction of efficient and accurate numerical methods for the solution of Volterra Integro-Differential Equations (VIDEs). In order to increase the order of convergence of classical one-step collocation methods, we propose multistep collocation methods, which
have been successfully introduced for Volterra integral equations in [1; 2]. Moreover, they are continuous
methods, i.e. they furnish an approximation of the solution at each point of the time interval. In this talk we describe the derivation of multistep collocation methods for VIDEs and the analysis of convergence and stability properties. We show some examples of methods which compare
favorably with respect to existing one-step methods. This is a joint work with B. Paternoster and D. Conte from University of Salerno.
REFERENCES
[1] D. Conte, Z. Jackiewicz, B. Paternoster. Two-step almost collocation methods for Volterra integral equations.
Appl. Math. Comput., 204 :839{853, 2008.
[2] D. Conte, B. Paternoster. Multistep collocation methods for Volterra Integral Equations. Appl. Math. Comput.,
59 :1721-1736, 2009
Multistep collocation methods for Volterra integro-differential equations
No full textMultistep collocation methods for Volterra integro–differential equations are derived and analyzed. They increase the order of convergence of classical one-step collocation methods, at the same computational cost. The numerical stability analysis is carried out and classes of A0-stable methods are provided. Numerical experiments confirm theoretical expectations and make comparisons with the one-step collocation methods
Efficient general linear methods for non-stiff differential equations
No full textThe aim of our research is the construction and analysis of efficient general linear methods (GLM), which achieve a good balance between accuracy and stability properties.
In order to reach our goal we consider the class of GLMs with quadratic stability (QS), i.e. methods whose stability function has only two non-zero roots [4]. This property simplifies the study of stability and the search for methods with high order and good stability properties. In this talk we describe the conditions which guarantee the QS property and the construction of explicit Nordsieck methods with QS and maximum area of the region of
absolute stability [2]. The search for these methods with high order is realized by various optimization routines [3], and the analogous search for another class of GLMs has been carried out in [1]. Examples of methods which compare favorably with respect to existing explicit GLM are presented, up to order six. Some issues concerning the implementation of our methods in a variable-step algorithm are addressed, such as the estimate of the
local error and the computation of the input vector for the next step. This is a joint work with G. Izzo from Università di Napoli 'Federico II' and Z. Jackiewicz from Arizona State University.
[1] M. Bras, A. Cardone, Construction of Efficient General Linear Methods for Non-Stiff Differential Systems, Math. Model. Anal. 17, 171-189 (2012).
[2] A. Cardone, Z. Jackiewicz, Explicit Nordsieck methods with quadratic stability, Numer. Algorithms, 60 (1), 1{25 (2012).
[3] A. Cardone, Z. Jackiewicz, H. D. Mittelmann, Optimization-based search for Nordsieck methods of high order with quadratic stability, to appear in Math. Model. Anal.
[4] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley, Hoboken, New Jersey, 2009
Numerical solution of time fractional diffusion systems
Full text linkIn this paper a general class of diffusion problem is considered, where the standard time derivative is replaced by a fractional one. For the numerical solution, a mixed method is proposed, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods to discretize the fractional derivative. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given
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