The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

We formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.     

Convergence Analysis of the Finite Difference ADI Scheme for Variable Coefficient Parabolic Problems with Nonzero Dirichlet Boundary Conditions

Computational Mathematics and Mathematical Physics - Since the invention by Peaceman and Rachford, more than 60 years ago, of the well celebrated ADI finite difference scheme for parabolic...     

Loss of benzene to generate an enolate anion by a site-specific double-hydrogen transfer during CID fragmentation of o-alkyl ethers of ortho-hydroxybenzoic acids

Collision-induced dissociation of anions derived from ortho-alkyloxybenzoic acids provides a facile way of producing gaseous enolate anions. The alkyloxyphenyl anion produced after an initial loss of CO(2) undergoes elimination of a benzene molecule by a double-hydrogen transfer mechanism, unique to the ortho isomer, to form an enolate anion. Deuterium labeling studies confirmed that the two hydrogen atoms transferred in the benzene loss originate from positions 1 and 2 of the alkyl chain. An initial transfer of a hydrogen atom from the C-1 position forms a phenyl anion and a carbonyl compound, both of which remain closely associated as an ion/neutral complex. The complex breaks either directly to give the phenyl anion by eliminating the neutral carbonyl compound, or to form an enolate anion by transferring a hydrogen atom from the C-2 position and eliminating a benzene molecule in the process. The pronounced primary kinetic isotope effect observed when a deuterium atom is transferred from the C-1 position, compared to the weak effect seen for the transfer from the C-2 position, indicates that the first transfer is the rate determining step. Quantum mechanical calculations showed that the neutral loss of benzene is a thermodynamically favorable process. Under the conditions used, only the spectra from ortho isomers showed peaks at m/z 77 for the phenyl anion and m/z 93 for the phenoxyl anion, in addition to that for the ortho-specific enolate anion. Under high collision energy, the ortho isomers also produce a peak at m/z 137 for an alkene loss. The spectra of meta and para compounds show a peak at m/z 92 for the distonic anion produced by the homolysis of the O--C bond. Moreover, a small peak at m/z 136 for a distonic anion originating from an alkyl radical loss allows the differentiation of para compounds from meta isomers. Copyright (c) 2008 John Wiley & Sons, Ltd.     

An unexpected ion-molecule adduct in negative-ion collision-induced decomposition ion-trap mass spectra of halogenated benzoic acids

The ion observed at m/z 145 when product ion spectra of iodobenzoate anions are recorded using ion-trap mass spectrometers corresponds to the adduct ion [I(H(2)O)](-). The elements of water required for the formation of this adduct do not originate from the precursor ion but from traces of moisture present in the helium buffer gas. A collision-induced decomposition (CID) spectrum recorded from the [M-H](-) ion (m/z 251) derived from 3-iodo[2,4,5,6-(2)H(4)]benzoic acid also showed an ion at m/z 145. This observation confirmed that the m/z 145 is not a product ion resulting from a direct neutral loss from the carboxylate anion. (79)Bromobenzoate anions produce similar results showing an ion at m/z 97 for [(79)Br(H(2)O)](-). The ion-molecule reaction observed here is unique to ion-trap mass spectrometers since a corresponding ion was not observed under our experimental conditions in spectra recorded with in-space tandem mass spectrometers such as triple quadrupole or quadrupole time-of-flight instruments.     

Matrix decomposition algorithms for elliptic boundary value problems: a survey

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N ² logN) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two–point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson’s equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems.     

ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.     

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N ² log₂ N + mN ²) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log₂ N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.      

Orthogonal spline collocation methods for biharmonic problems

Orthogonal spline collocation methods are formulated and analyzed for the solution of certain biharmonic problems in the unit square. Particular attention is given to the Dirichlet biharmonic problem which is solved using capacitance matrix techniques. Received November 11, 1996     

Multilevel additive and multiplicative methods for orthogonal spline collocation problems

Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Received March 1, 1994 / Revised version received January 23, 1996