The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform

Fabio Trani - One of the best experts on this subject based on the ideXlab platform.

  • Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations.
    Journal of Computational Chemistry, 2018
    Co-Authors: Domenico Ninno, Giovanni Cantele, Fabio Trani
    Abstract:

    We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators on discrete coordinate eigenkets |xn〉 defined on a uniform grid. Starting from the discretization of integrals involving canonical commutations, simple closed-form expressions of the matrix elements are obtained. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the derivative Approximation Order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method. © 2018 Wiley Periodicals, Inc.

  • Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations
    arXiv: Quantum Physics, 2017
    Co-Authors: Domenico Ninno, Giovanni Cantele, Fabio Trani
    Abstract:

    We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators using, as a basis set, the discrete coordinate eigenkets $\vert x_n\rangle$ defined on the uniform grid $x_n=na$. Simple closed form expressions of the matrix elements are obtained starting from integrals involving the canonical commutation rule. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the Approximation Order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method.

Bert Jüttler - One of the best experts on this subject based on the ideXlab platform.

  • c 2 hermite interpolation by pythagorean hodograph space curves
    Mathematics of Computation, 2007
    Co-Authors: Bert Jüttler
    Abstract:

    We solve the problem of C 2 Hermite interpolation by Pythagorean Hodograph (PH) space curves. More precisely, for any set of C 2 space boundary data (two points with associated first and second derivatives) we construct a four-dimensional family of PH interpolants of degree 9 and introduce a geometrically invariant parameterization of this family. This parameterization is used to identify a particular solution, which has the following properties. First, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Second, it has the best possible Approximation Order 6. Third, it is symmetric in the sense that the interpolant of the "reversed" set of boundary data is simply the "reversed" original interpolant. This particular PH interpolant is exploited for designing algorithms for converting (possibly piecewise) analytical curves into a piecewise PH curve of degree 9 which is globally C 2 , and for simple rational Approximation of pipe surfaces with a piecewise analytical spine curve. The algorithms are presented along with an analysis of their error and Approximation Order.

  • g 1 hermite interpolation by minkowski pythagorean hodograph cubics
    Computer Aided Geometric Design, 2006
    Co-Authors: Jiři Kosinka, Bert Jüttler
    Abstract:

    As observed by [Choi, H.I., Han, Ch.Y., Moon, H.P., Roh, K.H., Wee, N.S., 1999. Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Computer-Aided Design 31, 59-72], curves in Minkowski space R^2^,^1 are very well suited to describe the medial axis transform (MAT) of a planar domain, and Minkowski Pythagorean hodograph (MPH) curves correspond to domains, where both the boundaries and their offsets are rational curves [Moon, H.P., 1999. Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739-753]. Based on these earlier results, we give a thorough discussion of G^1 Hermite interpolation by MPH cubics, focusing on solvability and Approximation Order. Among other results, it is shown that any analytic space-like curve without isolated inflections can be approximately converted into a G^1 spline curve composed of MPH cubics with the Approximation Order being equal to four. The theoretical results are illustrated by several examples. In addition, we show how the curvature of a curve in Minkowski space is related to the boundaries of the associated planar domain.

  • approximating curves and their offsets using biarcs and pythagorean hodograph quintics
    Computer-aided Design, 2006
    Co-Authors: Zbynk Sir, Robert Feichtinger, Bert Jüttler
    Abstract:

    This paper compares two techniques for the Approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C^1 boundary data. We analyze the Approximation Order of both conversion procedures. As a new result, the C^1 Hermite interpolation with PH quintics is shown to have Approximation Order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher Approximation Order.

  • spatial pythagorean hodograph quintics and the Approximation of pipe surfaces
    Conference on Mathematics of Surfaces, 2005
    Co-Authors: Zbyněk Sir, Bert Jüttler
    Abstract:

    As observed by Farouki et al.[9], any set of C1 space boundary data (two points with associated first derivatives) can be interpolated by a Pythagorean hodograph (PH) curve of degree 5. In general there exists a two dimensional family of interpolants. In this paper we study the properties of this family in more detail. We introduce a geometrically invariant parameterization of the family of interpolants. This parameterization is used to identify a particular solution, which has the following properties. Firstly, it preserves planarity, i.e., the interpolant to planar data is a planar PH curve. Secondly, it has the best possible Approximation Order (4). Thirdly, it is symmetric in the sense that the interpolant of the “reversed” set of boundary data is simply the “reversed” original interpolant. These observations lead to a fast and precise algorithm for converting any (possibly piecewise) analytical curve into a piecewise PH curve of degree 5 which is globally C1. Finally we exploit the rational frames associated with any space PH curve (the Euler-Rodrigues frame) in Order to obtain a simple rational Approximation of pipe surfaces with a piecewise analytical spine curve and we analyze its Approximation Order.

Domenico Ninno - One of the best experts on this subject based on the ideXlab platform.

  • Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations.
    Journal of Computational Chemistry, 2018
    Co-Authors: Domenico Ninno, Giovanni Cantele, Fabio Trani
    Abstract:

    We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators on discrete coordinate eigenkets |xn〉 defined on a uniform grid. Starting from the discretization of integrals involving canonical commutations, simple closed-form expressions of the matrix elements are obtained. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the derivative Approximation Order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method. © 2018 Wiley Periodicals, Inc.

  • Real-space grid representation of momentum and kinetic energy operators for electronic structure calculations
    arXiv: Quantum Physics, 2017
    Co-Authors: Domenico Ninno, Giovanni Cantele, Fabio Trani
    Abstract:

    We show that the central finite difference formula for the first and the second derivative of a function can be derived, in the context of quantum mechanics, as matrix elements of the momentum and kinetic energy operators using, as a basis set, the discrete coordinate eigenkets $\vert x_n\rangle$ defined on the uniform grid $x_n=na$. Simple closed form expressions of the matrix elements are obtained starting from integrals involving the canonical commutation rule. A detailed analysis of the convergence toward the continuum limit with respect to both the grid spacing and the Approximation Order is presented. It is shown that the convergence from below of the eigenvalues in electronic structure calculations is an intrinsic feature of the finite difference method.

Hansjoachim Bungartz - One of the best experts on this subject based on the ideXlab platform.

  • quasi newton waveform iteration for partitioned surface coupled multiphysics applications
    International Journal for Numerical Methods in Engineering, 2020
    Co-Authors: Benjamin Ruth, Benjamin Uekermann, Miriam Mehl, Philipp Birken, Azahar Monge, Hansjoachim Bungartz
    Abstract:

    We present novel coupling schemes for partitioned multiphysics simulation that combine four important aspects for strongly coupled problems: implicit coupling per time step, fast and robust acceleration of the corresponding iterative coupling, support for multirate time stepping, and higher-Order convergence in time. To achieve this, we combine waveform relaxation—a known method to achieve higher-Order in applications with split time stepping based on continuous representations of coupling variables in time— with interface quasi-Newton coupling, which has been developed throughout the last decade and is generally accepted as a very robust iterative coupling method even for gluing together black-box simulation codes. We show convergence results (in terms of convergence of the iterative solver and in terms of Approximation Order in time) for two academic testcases—a heat transfer scenario and a fluid-structure interaction simulation. We show that we achieve the expected Approximation Order and that our iterative method is competitive in terms of iteration counts with those designed for simpler first-Order-in-time coupling. (Less)

  • quasi newton waveform iteration for partitioned fluid structure interaction
    arXiv: Numerical Analysis, 2020
    Co-Authors: Benjamin Ruth, Benjamin Uekermann, Miriam Mehl, Philipp Birken, Azahar Monge, Hansjoachim Bungartz
    Abstract:

    We present novel coupling schemes for partitioned multi-physics simulation that combine four important aspects for strongly coupled problems: implicit coupling per time step, fast and robust acceleration of the corresponding iterative coupling, support for multi-rate time stepping, and higher-Order convergence in time. To achieve this, we combine waveform relaxation -- a known method to achieve higher Order in applications with split time stepping based on continuous representations of coupling variables in time -- with interface quasi-Newton coupling, which has been developed throughout the last decade and is generally accepted as a very robust iterative coupling method even for gluing together black-box simulation codes. We show convergence results (in terms of convergence of the iterative solver and in terms of Approximation Order in time) for two academic test cases -- a heat transfer scenario and a fluid-structure interaction simulation. We show that we achieve the expected Approximation Order and that our iterative method is competitive in terms of iteration counts with those designed for simpler first-Order-in-time coupling.

Xuan Zhao - One of the best experts on this subject based on the ideXlab platform.

  • compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions
    Journal of Computational Physics, 2013
    Co-Authors: Xuan Zhao
    Abstract:

    An effective finite difference scheme is considered for solving the time fractional sub-diffusion equation with Neumann boundary conditions. A difference scheme combining the compact difference approach the spatial discretization and L"1 Approximation for the Caputo fractional derivative is proposed and analyzed. Although the spatial Approximation Order at the Neumann boundary is one Order lower than that for interior mesh points, the unconditional stability and the global convergence Order O(@t^2^-^@a+h^4) in discrete L"2 norm of the compact difference scheme are proved rigorously, where @t is the temporal grid size and h is the spatial grid size. Numerical experiments are included to support the theoretical results, and comparison with the related works are presented to show the effectiveness of our method.