One-dimensional Model Problem Research Articles

Multibody scattering, correlation, molecular conduction, and the 0.7 anomaly

We describe a new grid-based (or localized orbital-based) method for treating the effects of exchange and correlation on electronic transmission through a molecular target where there are initially other bound electrons. Our algorithm combines the approaches of (i) solid-state grid-based algorithms using self-energies and (ii) the complex Kohn method from electron-molecule scattering. For the general problem of a molecular target with n-electrons, our algorithm should ideally solve for electronic transmission with a computational cost scaling as n(2), although the present implementation is limited to one-dimensional problems. In this paper, we implement our algorithm to solve three one-dimensional model problems involving two electrons: (i) Single-channel resonant transmission through a double-barrier well (DBW), where the target already contains one bound-state electron [Rejec et al., Phys. Rev. B 67, 075311 (2003)]; (ii) multichannel resonant transmission through a DBW, where the incoming electron can exchange energy with the bound electron; (iii) transmission through a triple-barrier well (TBW), where the incoming electron can knock forward the bound electron, yielding a physical model of electron-assisted electron transfer. This article offers some insight about the role and size of exchange and correlation effects in molecular conduction, where few such rigorous calculations have yet been made. Such multibody effects have already been experimentally identified in mesoscopic electron transport, giving rise to the "0.7 anomaly," whereby electrons traveling through a narrow channel pair up as singlets and triplets. We expect the effect of electronic correlation to be even more visible for conduction through molecules, where electrons should partially localize into bonding and antibonding orbitals.

Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics

The dynamics of gaseous stars is often described by magnetic fields coupled to self-gravitation and radiation effects. In this paper we consider an initial–boundary value problem for nonlinear planar magnetohydrodynamics (MHD) in the case that the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes are taken into account. Based on the fundamental local existence results and global-in-time a priori estimates, we establish the global existence of a unique classical solution with large initial data to the initial–boundary value problem under quite general assumptions on the heat conductivity.

Stabilization of resonance states by an asymptotic Coulomb potential

The capability of using stabilization method with the asymptotic Coulomb potential for accurate resonance state parameter calculations is investigated for the one-dimensional model problem with a single open channel. It is shown that the replacement of the potential box conditions by the quantization with asymptotic Coulomb potential does not yield the significant errors in energy and width of resonances if the additional mathematical conditions are correctly taken into account. Analytical results are confirmed by the numerical calculations.

An adaptive high-order discontinuous Galerkin method with error control for the Hamilton–Jacobi equations. Part I: The one-dimensional steady state case

We propose and study an adaptive version of the discontinuous Galerkin method for Hamilton–Jacobi equations. It works as follows. Given the tolerance and the degree of the polynomial of the approximate solution, the adaptive algorithm finds a mesh on which the approximate solution has an L ∞-distance to the viscosity solution no bigger than the prescribed tolerance. The algorithm uses three main tools. The first is an iterative solver combining the explicit Runge–Kutta discontinuous Galerkin method and the implicit Newton’s method that enables us to solve the Hamilton–Jacobi equations efficiently. The second is a new a posteriori error estimate based on the approximate resolution of an approximate problem for the actual error. The third is a method that allows us to find a new mesh as a function of the old mesh and the ratio of the a posteriori error estimate to the tolerance. We display extensive numerical evidence that indicates that, for any given polynomial degree, the method achieves its goal with optimal complexity independently of the tolerance. This is done in the framework of one-dimensional steady-state model problems with periodic boundary conditions.

Symmetric low-frequency motion in incompressible transversely isotropic elastic plates

The problem of long-wave low-frequency extensional (symmetric) motion in a layer composed of incompressible, transversely isotropic elastic material is investigated. Motivated by appropriate approximations of the dispersion relation, a hierarchy of asymptotically approximate boundary value problems is set up and solved. A leading order system of equations is obtained for the governing extensions, together with a refined system for their second order counterparts. A one-dimensional model problem, involving impact edge loading, is set up and solved in order to illustrate the derived theory.

Selecting the Numerical Flux in Discontinuous Galerkin Methods for Diffusion Problems

In this paper we present numerical investigations of four different formulations of the discontinuous Galerkin method for diffusion problems. Our focus is to determine, through numerical experimentation, practical guidelines as to which numerical flux choice should be used when applying discontinuous Galerkin methods to such problems. We examine first an inconsistent and weakly unstable scheme analyzed in Zhang and Shu, Math. Models Meth. Appl. Sci. (M3 AS) 13, 395-413 (2003), and then proceed to examine three consistent and stable schemes: the Bassi-Rebay scheme (J. Comput. Phys. 131, 267 (1997)), the local discontinuous Galerkin scheme (SLAM J. Numer. Anal. 35, 2440-2463 (1998)) and the Baumann-Oden scheme (Comput. Math. Appl. Mech. Eng. 175, 311-341 (1999)). For an one-dimensional model problem, we examine the stencil width, h-convergence properties, p-convergence properties, eigenspectra and system conditioning when different flux choices are applied. We also examine the ramifications of adding stabilization to these schemes. We conclude by providing the pros and cons of the different flux choices based upon our numerical experiments.

A comparison of several numerical methods for the solution of the Convection-Diffusion Equation using the method of finite spheres

In this paper we compare several numerical methods for the solution of the convection-diffusion equation using the method of finite spheres; a truly meshfree numerical technique for the solution of boundary value problems. By conducting numerical inf-sup tests on a one-dimensional model problem it is found that a higher order derivative artificial diffusion (Ho DAD) method performs the best among the schemes tested. This method is then applied to the analysis of problems in two-dimensions.

A multigrid preconditioner and automatic differentiation for non-equilibrium radiation diffusion problems

We study the efficient solution of non-equilibrium radiation diffusion problems. An implicit time discretization leads to the solution of systems of non-linear equations which couple radiation energy and material temperature. We consider the implicit Euler method, the mid-point scheme, the two-step backward differentiation formula, and a two-stage implicit Runge–Kutta method for time discretization. We employ a Newton–Krylov method in the solution of arising non-linear problems. We describe the computation of the Jacobian matrix for Newton’s method using automatic differentiation based on the operator overloading in Fortran 90. For GMRES iterations, we propose a simple multigrid preconditioner applied directly to the coupled linearized problems. We demonstrate the efficiency and scalability of the proposed solution procedure by solving one-dimensional and two-dimensional model problems.

Non-reflecting boundary conditions for atomistic, continuum and coupled atomistic/continuum simulations

We present a method to numerically calculate a non-reflecting boundary condition which is applicable to atomistic, continuum and coupled multiscale atomistic/continuum simulations. The method is based on the assumption that the forces near the domain boundary can be well represented as a linear function of the displacements, and utilizes standard Laplace and Fourier transform techniques to eliminate the unnecessary degrees of freedom. The eliminated degrees of freedom are accounted for in a time-history kernel that can be calculated for arbitrary crystal lattices and interatomic potentials, or regular finite element meshes using an automated numerical procedure. The new theoretical developments presented in this work allow the application of the method to non-nearest neighbour atomic interactions; it is also demonstrated that the identical procedure can be used for finite element and mesh-free simulations. We illustrate the effectiveness of the method on a one-dimensional model problem, and calculate the time-history kernel for FCC gold using the embedded atom method (EAM). Copyright © 2005 John Wiley & Sons, Ltd.

An adaptive method with rigorous error control for the Hamilton–Jacobi equations. Part I: The one-dimensional steady state case

In this paper, we introduce a new adaptive method for finding approximations for Hamilton–Jacobi equations whose L ∞ -distance to the viscosity solution is no bigger than a prescribed tolerance. This is done on the simple setting of a one-dimensional model problem with periodic boundary conditions. We consider this to be a stepping stone towards the more challenging goal of constructing such methods for general Hamilton–Jacobi equations. The method proceeds as follows. On any given grid, the approximate solution is computed by using a well-known monotone scheme; then, the quality of the approximation is tested by using an approximate a posteriori error estimate. If the error is bigger than the prescribed tolerance, a new grid is computed by solving a differential equation whose devising is the main contribution of the paper. A thorough numerical study of the method is performed which shows that rigorous error control is achieved, even though only an approximate a posteriori error estimate is used; the method is thus reliable. Furthermore, the numerical study also shows that the method is efficient and that it has an optimal computational complexity. These properties are independent of the value of the tolerance. Finally, we provide extensive numerical evidence indicating that the adaptive method converges to an approximate solution that can be characterized solely in terms of the tolerance, the artificial viscosity of the monotone scheme and the exact solution.

On resonances in open systems

An open system or open resonator is a domain of wave activity separated from the exterior by a partly open or partly transparent surface. Such open resonators lose energy to infinity through radiation. The numerical computation of the corresponding resonances is complicated by spurious reflections of the outgoing waves at the necessarily finite grid boundaries. These reflections can be reduced to extremely low levels by applying perfectly matched layer (PML) absorbing boundary conditions, which separate the discrete resonances from the continuous spectrum. Using a simple one-dimensional model problem, the influence of the various PML parameters is determined by a numerical error analysis. In addition to one-dimensional open resonators, two-dimensional open resonators as well as various resonating structures in waveguides are considered, and the resonant spectra and selected modes are evaluated. For the first time, leaky modes are computed for several resonating structures in a waveguide in addition to the trapped modes published in the literature. In applications, leaky mode resonances are often more important than trapped mode resonances. Gap tones, observed in a model problem of high-lift configurations, are identified as transversal resonant modes with the lowest radiation losses.

Coupled chemo-mechanical deterioration of cementitious materials Part II: Numerical methods and simulations

The paper is concerned with numerical solution procedures for coupled chemo-mechanical simulations of concrete structures subjected to calcium leaching and mechanical damage. The analyses are based upon the chemo-mechanical damage model for cementitious materials presented in the first part of the paper [Int. J. Solids Struct., in press]. These analyses are characterized by a non-linear transient two-field problem described by the displacement field, the pore fluid calcium concentration field and its rate as state variables. The related non-linear first order initial boundary value problem is solved numerically by a semi-discretization strategy. For the temporal discretization the second order accurate implicit generalized mid-point rule is used. The spatial discretization is based upon hierarchical finite elements. The non-linear algebraic system of equations is solved by consistent linearization and Newton–Raphson iteration. Representative simulations of calcium leaching and the coupled chemo-mechanical degradation of concrete structures are presented. In particular, the process of calcium leaching is studied in detail by means of a one-dimensional model problem. Chemo-mechanical interactions on the structural level are investigated in the numerical simulation of a concrete beam subjected simultaneously to mechanical loads and deionized water. It is demonstrated that chemical damage not only reduces the limit load and the stiffness of concrete structures but also their structural behavior.

A monolithic approach to fluid–structure interaction

This paper compares partitioned and monolithic solution procedures for the numerical simulation of fluid–structure interactions. Their different stability properties are illustrated and the role of structural prediction for a partitioned method is discussed. A grid refinement study has been carried out to assess the temporal accuracy of these methods. Moreover, their computational cost as well as their computational efficiency is compared. Numerical experiments are presented for a one-dimensional model problem of a piston interacting with a fluid.

Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem

We study a one dimensional Dirichlet problem of fourth order and a corresponding "buckling eigenvalue problem" under Dirichlet boundary conditions. These problems may serve as model problems for "Orr-Sommerfeld" like boundary and eigenvalue problems. It turns out that eigenvalue curves in appropriate parameter domains look completely different than for the same equation under so called Navier boundary conditions. Further emphasis is laid on positivity properties, and also here, fundamental differences with Navier conditions arise: It may e.g. happen that one has infinitely many linearly independent positive eigenfunctions. Connections and analogies with the clamped plate boundary value problem on families of deformed domains are discussed.

Comparison of numerical schemes for the solution of the advective age equation in ice sheets

AbstractA one-dimensional model problem for computation of the age field in ice sheets, which is of great importance for dating deep ice cores, is considered. the corresponding partial differential equation (PDE) is of purely advective (hyperbolic) type, which is notoriously difficult to solve numerically. By integrating the PDE over a space–time element in the sense of a finite-volume approach, a general difference equation is constructed from which a hierarchy of solution schemes can be derived. Iteration rules are given explicitly for central differences, first-, second- and third-order (QUICK) upstreaming as well as modified TVD Lax–Friedrichs schemes (TVDLFs). the performance of these schemes in terms of convergence and accuracy is discussed. Second-order upstreaming, themodifiedTVDLF scheme with Minmod slope limiter and, with limitations of the accuracy directly at the base, first-order upstreaming prove to be the most suitable for numerical age computations in ice-sheet models.

A residual a posteriori error estimator for the finite element solution of the Helmholtz equation

This paper suggests an expression of a residual-based a posteriori error estimation for a Galerkin finite element discretisation of the Helmholtz operator applied in acoustics. In the first part, we summarise the finite element approach and the a priori estimates. The new residual estimator is then formulated, illustrated and tested on two one-dimensional model problems with closed form solution.

A Multiscale Pressure Splitting of the Shallow-Water Equations: I. Formulation and 1D Tests

Direct representation of the free surface in ocean circulation models leads to a number of computational difficulties that are due to the fast time scales associated with free-surface waves. These fast time scales generally result in severe time-step restrictions when the free surface is advanced using an explicit scheme and may result in large phase errors when the free surface is treated implicitly with a large time step. A multiple-scale analysis of the shallow-water equations is used to analyze this stiffness and to guide the construction of a computational methodology that overcomes the associated difficulties. Specifically, we explore a class of fractional step methods that utilize coarsened grids in the propagation of long-wave data. The behavior of the corresponding schemes is examined in detail in light of one-dimensional model problems, based on finite-difference or spectral-element discretizations.

On stabilized finite element methods for linear systems of convection–diffusion-reaction equations

A stabilized finite element method for solving systems of convection–diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a one-dimensional (1D) model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner–Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.

Simulation of nonadiabatic wave packet interferometry using classical trajectories

In this paper, we describe the application of our recently developed multistate semiclassical Liouville equation method for modeling molecular dynamics on multiple coupled electronic states [C. C. Martens and J.-Y. Fang, J. Chem. Phys. 106, 4918 (1997); A. Donoso and C. C. Martens, J. Phys. Chem. 102, 4291 (1998)] to problems where electronic coherence effects play a dominant role. We consider a model problem involving the simultaneous evolution of wave packets on two coupled electronic states. We analyze the problem qualitatively from both quantum and semiclassical perspectives using perturbation theory, and identify the roles played by coupling strength and relative phase of the initial wave packets. We then perform trajectory-based simulations on a two-state one-dimensional model problem and compare the results with those of exact quantum calculations. In marked contrast with most current methods for modeling nonadiabatic dynamics with classical trajectories, the semiclassical Liouville method is found to be capable of treating even dominant electronic coherence effects in a consistent and accurate manner.

Two-phase modeling of mushy zone parameters associated with hot tearing

A two-phase continuum model for an isotropic mushy zone is presented. The model is based upon the general volume-averaged conservation equations, and quantities associated with hot tearing are included, i.e., after-feeding of the liquid melt due to solidification shrinkage is taken into account as well as thermally induced deformation of the solid phase. The model is implemented numerically for a one-dimensional model problem with some similarities to the aluminium direct chill (DC) casting process. The variation of some key parameters that are known to influence the hot-tearing tendency is then studied. The results indicate that both liquid pressure drop due to feeding difficulties and tensile stress caused by thermal contraction of the solid phase are necessary for the formation of hot tears. Based upon results from the one-dimensional model, it is furthermore concluded that none of the hot-tearing criteria suggested in the literature are able to predict the variation in hot-tearing susceptibility resulting from a variation in all of the following parameters: solidification interval, cooling contraction of the solid phase, casting speed, and liquid fraction at coherency.