Ernest Mund

Chebyshev pseudospectral solution of second-order elliptic equations with finite element preconditioning

Abstract A new Chebyshev pseudospectral algorithm for second-order elliptic equations using finite element preconditioning is proposed and tested on various problems. Bilinear and biquadratic Lagrange elements are considered as well as bicubic Hermite elements. The numerical results show that bilinear elements produce spectral accuracy with the minimum computational work. L -shaped regions are treated by a subdomain approach.

Finite-element preconditioning for pseudospectral solutions of elliptic problems

A preconditioning technique for pseudospectral solutions of elliptic problems based on quadrangular finite-element algorithms is analyzed, which exhibits excellent convergence properties. The pseudospectral technique is implemented through a collocation grid based on Gauss–Lobatto quadrature nodes associated to the Jacobi orthogonal polynomials. Various types of basis functions are used in the finite-element preconditioner (i.e., low-order Lagrange or cubic Hermite elements). Dirichlet and Neumann problems are investigated in one- and two-space dimensions. Numerical results show that the eigenvalue spectrum of the iteration matrix is inside the unit circle and even, close to zero for a wide range of operators. This property ensures convergence until roundoff error level in a few iterations. The differences between finite-element and finite-difference preconditioning are analyzed. Finally, the application of the algorithm to a problem exhibiting geometric induced singularities is discussed.

Fourier analysis of finite element preconditioned collocation schemes

This paper investigates the spectrum of the iteration operator of some finite element preconditioned Fourier collocation schemes. The first part of the paper analyses one-dimensional elliptic and hyperbolic model problems and the advection-diffusion equation. Analytical expressions of the eigenvalues are obtained with use of symbolic computation. The second part of the paper considers the set of one-dimensional differential equations resulting from Fourier analysis (in the transverse direction) of the two-dimensional Stokes problem. All results agree with previous conclusions on the numerical efficiency of finite element preconditioning schemes.

On the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation calculations

Abstract An analytical investigation is performed of the spectrum of the iteration operator associated to the finite element preconditioning of Chebyshev collocation calculations, on a one-dimensional Dirichlet model problem. Use is made of the techniques developed by Haldenwang et al. for the finite difference preconditioning of the same problem. In the latter case the eigenvalues may be obtained as the diagonal elements of an upper-triangular matrix. This is not possible with finite element preconditioning, where the corresponding operator splits into two parts, one of which only is upper-triangular. Discarding the non-triangular part of the operator (which vanishes asymptotically for large values of N , partition size), this procedure yields an approximate expression of the eigenvalues in good agreement with the properties given earlier by Deville and Mund.

On a Stable Spectral Method for the grad(div) Eigenvalue Problem

We present in this paper a stable spectral element for the approximations of the grad(div) eigenvalue problem in two and three-dimensional quadrangular geometry. Spectral approximations based on Gaussian quadrature rules are built in a dual variational approach with Darcy type equations. We prove that spectral convergence can be reached for the irrotational spectrum without the presence of any spurious eigenmodes, provided an adequate choice is made for the quadrature rules.

Preconditioned Chebyshev collocation methods and triangular finite elements

This paper analyzes triangular finite elements for the preconditioning of Chebyshev collocation solutions of elliptic boundary value problems. Results are given for scalar model problems and for both Stokes and Navier-Stokes equations.

Numerical solution of neutron kinetics equations using A-stable algorithms

This paper gives a detailed account of both theoretical and numerical investigations which have been conducted in the application of A-stable algorithms to neutron kinetics problems. It is broadly divided into three sections. General considerations on desirable features of a reactor dynamics code are followed by the theoretical background. In order to be self-contained, the stability properties of one-step methods are recalled with emphasis on the A-stability concept introduced by Dahlquist. An algorithm is described, based on the interpolation of exp(z) in the unit disc of the complex plane, which generates A-stable schemes wnn(z), (n= 1,…) with so-called ‘spectral matching’ properties. Practical reasons limit to w11 (z) its use for the integration of the kinetics equations and the analytical properties of this first order rational approximation to the exponential function are studied. A second class of suitable integration schemes is made of the implicit Runge-Kutta (IRK) family, particularly the subclass of diagonally implicit Runge-Kutta (DIRK) methods which are factorizable. Finally, the numerical results obtained with these algorithms are discussed on a set of four point kinetics problems for both fast and thermal-type reactors.

RTk/SN solutions of the two-dimensional multigroup transport equations in hexagonal geometry

Abstract This paper describes an extension to the hexagonal geometry of some weakly discontinuous nodal finite element schemes developed by Hennart and del Valle for the two-dimensional discrete ordinates transport equation in quadrangular geometry. The extension is carried out in a way similar to the extension to the hexagonal geometry of nodal element schemes for the diffusion equation using a composite mapping technique suggested by Hennart, Mund, and del Valle. The combination of the weakly discontinuous nodal transport scheme and the composite mapping is new and is detailed in the main section of the paper. The algorithm efficiency is shown numerically through some benchmark calculations on classical problems widely referred to in the literature.

A short survey on preconditioning techniques in spectral calculations

Two decades ago collocation methods for high accuracy solutions of partial differential equations seemed applicable only to special classes of problems with simple geometries, regular solutions, absence of numerical round-off errors, etc. The usefulness of these methods in engineering problems was very much in question. The situation has changed with many obstacles having been removed which prevented spectral methods to be used with maximum performance. We briefly review a key element in this change: the use of preconditioning techniques

Calculating the Discrete Spectrum of the Transport Operator with Arbitrary Order Anisotropic Scattering

We consider here the numerical determination of the discrete spectrum of the neutron transport operator in a homogeneous subcritical infinite medium in the case of highly anisotropic scattering. Under mild assumptions on the scattering coefficients, one knows that the discrete spectrum is a finite set whose elements are real with absolute values larger than 1. Under these assumptions, we describe a method to compute all eigenvalues for an arbitrary order of anisotropic scattering. The method has been implemented in a C++ library using IEEE double precision arithmetic, allowing for fast calculations. Numerical results are compared to values in literature and to high‐precision calculations using Maple.