This paper describes a simple but effective technique for reducing dispersion errors in finite element solutions of timeharmonic wave propagation problems. The method involves a simple shift of the integration points to locations away from conventional Gauss or Gauss–Lobatto integration points. For bilinear rectangular elements, such a shift results in fourth-order accuracy with respect to disp...

Abstract. This paper pr esents an energ y-optimal trajectory planning method fo r spacecraft fo rmation reconfiguration in deep space environment using continuous lo w-thrust propulsion system. First, we emplo y the Legendre pseudospectral method (LPM) to transform the optimal reconfiguration problem to a parameter optimization nonlinear programming (NLP) problem. Then, to avoid the computation...

In this paper, a numerical method is proposed for solving optimal control problem of Volterra integral equations using radial basis functions (RBFs) for approximating unknown function. Actually, the method is based on interpolation by radial basis functions including multiquadrics (MQs), to determine the control vector and the corresponding state vector in linear dynamic system while minimizing...

The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point values (or' an highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponen...

The cornerstone of nodal spectral element methods is the co-location of the interpolation and integration points, yielding a diagonal mass matrix that is efficient for time-integration methods. On quadrilateral elements Legendre-Gauss-Lobatto points are both good interpolation and integration points but on triangles analogous points have not yet been found. In this paper we use a promising set ...

We describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. Our focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. Our algorithm com...

It is shown that as m tends to infinity, the error in the integration of the Chebyshev polynomial of the first kind, T{im+2)j±2^x), by an /n-point Gauss integration rule approaches (-!> • 2/(4/2 1), / = 0, 1, ■ • • , m 1, and (-!>' • tt/2, / = m, for all J. 1. Knowledge of the errors in the numerical integration of Chebyshev polynomials of the first kind, Tn(x), by given integration rules has p...

In this paper, we develop a Jacobi-Gauss-Lobatto collocation method for solving the nonlinear fractional Langevin equation with three-point boundary conditions. The fractional derivative is described in the Caputo sense. The shifted Jacobi-Gauss-Lobatto points are used as collocation nodes. The main characteristic behind the Jacobi-Gauss-Lobatto collocation approach is that it reduces such a pr...

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[f ] when f(x) has general algebraic-logarithmic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the forms f(x) ∼ ∞ ∑ ...

In this paper, we examine the influence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss-Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments. c © ??? John Wile...