22 research outputs found
Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations
A construction of Marsden’s identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that repro-duces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Effi-cient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström’s method is used to numericallysolve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are pre-sented.Universidad de Granada / CBU
Solving boundary value problems via the Nyström method using spline Gauss rules
We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts
Solving Boundary Value Problems Via the Nyström Method Using Spline Gauss Rules
We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves a use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts
Generalized extrapolation methods for solving nonlinear Fredholm integral equations
In this paper we develop a class of generalized extrapolation methods for numerical solution of nonlinear Fredholm integral equations of the second kind. The direct representation of this class allows us to simply discretize the nonlinear integral equations with smooth kernels. This approach enjoys several outstanding features of numerical methods such as: economized computational cost, the high order accuracy, direct implementation, discrretization on arbitrary nodes and applying the methods with positive weights. The comparison results demonstrate the superior results of the new class of methods versus the classical and recent approaches
A collocation IGA-BEM for 3D potential problems on unbounded domains
In this paper the numerical solution of potential problems defined on 3D
unbounded domains is addressed with Boundary Element Methods (BEMs), since in
this way the problem is studied only on the boundary, and thus any finite
approximation of the infinite domain can be avoided. The isogeometric analysis
(IGA) setting is considered and in particular B-splines and NURBS functions are
taken into account. In order to exploit all the possible benefits from using
spline spaces, an important point is the development of specific cubature
formulas for weakly and nearly singular integrals. Our proposal for this aim is
based on spline quasi-interpolation and on the use of a spline product formula.
Besides that, a robust singularity extraction procedure is introduced as a
preliminary step and an efficient function-by-function assembly phase is
adopted. A selection of numerical examples confirms that the numerical
solutions reach the expected convergence orders.Comment: 17 pages, 4 figure
Excitations and spectra from equilibrium real-time Green's functions
The real-time contour formalism for Green's functions provides time-dependent
information of quantum many-body systems. In practice, the long-time simulation
of systems with a wide range of energy scales is challenging due to both the
storage requirements of the discretized Green's function and the computational
cost of solving the Dyson equation. In this manuscript, we apply a real-time
discretization based on a piece-wise high-order orthogonal-polynomial expansion
to address these issues. We present a superconvergent algorithm for solving the
real-time equilibrium Dyson equation using the Legendre spectral method and the
recursive algorithm for Legendre convolution. We show that the compact high
order discretization in combination with our Dyson solver enables long-time
simulations using far fewer discretization points than needed in conventional
multistep methods. As a proof of concept, we compute the molecular spectral
functions of H, LiH, He and CHO using self-consistent
second-order perturbation theory and compare the results with standard quantum
chemistry methods as well as the auxiliary second-order Green's function
perturbation theory method