1,721,008 research outputs found
On the numerical computation of the LMM's coefficients
No full textIn this work some remarks on the numerical computation of the coefficients of some families of LMMs, used with boundary conditions and on general meshes, are introduced. Three formulations of the order conditions, which are the tools used for determining such coefficients, are considered. In particular, the attention is focused on the Generalized BDF (GBDF) for which some experiments performed in MATLAB are also presented
Correction to: "On a class of Hermite-Obreshkov one-step methods with continuous spline extension" [Axioms 7 (3), 58, 2018]
Full text linkThe authors of the above mentioned paper specify that the considered class of one-step symmetric Hermite-Obreshkov methods satisfies the property of conjugate-symplecticity up to order p + r, where r = 2 and p is the order of the method. This generalization of conjugate-symplecticity states that the methods conserve quadratic first integrals and the Hamiltonian function over time intervals of length O(h-r). Theorem 1 of the above mentioned paper is then replaced by a new one. All the other results in the paper do not change. Two new figures related to the already considered Kepler problem are also added
Numerical aspects of the coefficient computation for LMMs
No full textThe numerical solution of Boundary Value Problems usually requires the use of an adaptive mesh selection strategy. For this reason, when a Linear Multistep Method is considered, a dynamic computation of its coefficients is necessary. This leads to solve linear systems which can be expressed in different forms, depending on the polynomial basis used to impose the order conditions. In this paper, we compare the accuracy of the numerically computed coefficients for three different formulations. For all the considered cases Vandermonde systems on general abscissae are involved and they are always solved by the Bj ̈rk-Pereyra algorithm. An adaptation of the forward error analysis given in [8, 9] is proposed whose significance is confirmed by the numerical results
The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions
No full textThe BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class of Boundary Value Methods for the solution of Ordinary Differential Equations. In the ODE context, using the numerical solution and the associated numerical derivative produced by the BS methods, it is possible to compute, with a local approach, a suitable spline with knots at the mesh points collocating the differential equation at the knots and having the same convergence order as the numerical solution. Starting from this spline, here we derive a new quasi-interpolation scheme having the function and the derivative values at the knots as input data. When the knot distribution is uniform or the degree is low, explicit formulas can be given for the coefficients of the new quasi-interpolant in the B-spline basis. In the general case these coefficients are obtained as solution of suitable local linear systems of size 2d×2d, where d is the degree of the spline. The approximation order of the presented scheme is optimal and the numerical results prove that its performances can be very good, in particular when suitable knot distributions are used
Quadrature formulas descending from BS Hermite spline quasi-interpolation
No full textTwo new classes of quadrature formulas associated to the BS Boundary Value Methods are discussed. The first is of Lagrange type and is obtained by directly applying the BS methods to the integration problem formulated as a (special) Cauchy problem. The second descends from the related BS Hermite quasi-interpolation approach which produces a spline approximant from Hermite data assigned on meshes with general distributions. The second class formulas is also combined with suitable finite difference approximations of the necessary derivative values in order to define corresponding Lagrange type formulas with the same accuracy. (C) 2012 Elsevier B.V. All rights reserved
Rational rotation–minimizing polar oriented rigid body motions
No full textIn this work we propose a scheme for defining a rational curve in the space, with an
associated rational rotation–minimizing directed frame. These curves are useful, for instance, when
we want to describe the motion of a rigid body having a main axis pointing always to a fixed point
and they can be employed in many interesting applications like camera motion control and medical
endoscopy. The construction entails interpolation of initial/final curve positions and directions, together
with the associated end frame orientations. The effectiveness of the method is illustrated by
some examples
BS methods: A new class of spline collocation BVMs
No full textBS methods are a recently introduced class of Boundary Value Methods which is based on B-splines. They can also be interpreted as spline collocation methods. For uniform meshes, the coefficients defining the k-step BS method are just the values of the (k+1)-degree uniform B-spline and B-spline derivative at its integer active knots; for general nonuniform meshes they are computed by solving local linear systems whose dimension depends on k. An important specific feature of BS methods is the possibility to associate a spline of degree k +1 and smoothness Ck to the numerical solution produced by the k-step method of this class. Such spline collocates the differential equation at the knots, shares the convergence order with the numerical solution, and can be computed with negligible additional computational cost. Here a survey on such methods is given, presenting the general definition, the convergence and stability features, and introducing the strategy for the computation of the coefficients in the B-spline basis which define the associated spline. Finally, some related numerical results are also presented. © 2008 American Institute of Physics
The continuous extension of the B-spline linear multistep methods for BVPs on non-uniform meshes
No full textB-spline methods are Linear Multistep Methods based on B-splines which have good stability properties [F. Mazzia, A. Sestini, D. Trigiante, B-spline multistep methods and their continuous extensions, SIAM J. Numer Anal. 44 (5) (2006) 1954-1973] when used as Boundary Value Methods [L. Brugnano, D. Trigiante, Convergence and stability of boundary value methods for ordinary differential equations, J. Comput. Appl. Math. 66 (1-2) (1996) 97-109; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach Science Publishers, Amsterdam, 1998]. In addition, they have an important feature: if k is the number of steps, it is always possible to associate to the numerical solution a Ck spline of degree k + 1 collocating the differential equation at the mesh points. In this paper we introduce an efficient algorithm to compute this continuous extension in the general case of a non-uniform mesh and we prove that the spline shares the convergence order with the numerical solution. Some numerical results for boundary value problems are presented in order to show that the use of the information given by the continuous extension in the mesh selection strategy and in the Newton iteration makes more robust and efficient a Matlab code for the solution of BVPs
B-spline linear multistep methods and their continuous extensions
No full textIn this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) with collocation points coinciding with the knots. Such collocation methods are naturally associated to a special class of linear multistep methods, here called B-spline (BS) methods, which are able to generate the spline values at the knots. We prove that, provided the additional conditions are appropriately chosen, such methods are all convergent and A-stable. The convergence property of the BS methods is naturally inherited by the related spline extensions, which, by the way, are easily and safely computable using their B-spline representation
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