We address quadrature-based approximations of the bilinear inverse form u>A−1u, where A is a real symmetric positive definite matrix, and analyze properties of the Gauss, Gauss-Radau, and Gauss-Lobatto quadrature. In particular, we establish monotonicity of the bounds given by these quadrature rules, compare the tightness of these bounds, and derive associated convergence rates. To our knowledg...

A numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method is presented in this article. A Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the implicit RungeKutta scheme are employed to obtain highly accurate approximations to the mentioned problem. The collocation points are the Chebyshev interpolation n...

We extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkinmethod while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse. The algorithm can be implemented efficiently and i...

A simple strategy for constructing a sequence of increasingly refined interpolation grids over the triangle or the tetrahedron is discussed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The interpolation nodes are generated based on a one-dimensional master grid comprised of the zeros of the Lobatto, Legendre, Chebyshev, and second-kind Chebyshev polyn...

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several ...

The problem of crack propagation along the interface of two bonded dissimilar orthotropic plates is considered. Using Galilean transformation, the problem is reduced to a quasistatic one. Then, using Fourier transforms and asymptotic analysis, the problem is reduced to a pair of singular integral equations with Cauchy-type singularity. These equations are solved using Gauss-Chebyshev quadrature...

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...

Recently, the Legendre pseudospectral (PS) method migrated from theory to flight application onboard the International Space Station for performing a finite-horizon, zeropropellant maneuver. A small technical modification to the Legendre PS method is necessary to manage the limiting conditions at infinity for infinite-horizon optimal control problems. Motivated by these technicalities, the conc...