bluuebunny/crossref_metadata_2025_split huggingface dataset:
Unified approach to numerical methods. III. Finite differences and ordinary differential equations
Authors: Herrera Ismael , Chargoy Lucia , Alduncin Gonzalo 
Published: August 2005

AbstractThis is the third of a sequence of articles devoted to the presentation of a direct method of analysis recently developed by one of the authors. The approach is quite general, since it is applicable to any linear operator, symmetric or nonsymmetric, regardless of its type. In particular, the theory includes steady‐state and time‐dependent problems. In this article the method is applied to ordinary differential equations, which constitute a very convenient illustration of its application because the analysis can be carried out in an exhaustive manner. General algorithms which yield the exact values of the solution or its derivatives are obtained in this manner.
Variable trial functions and the CVBEM
Authors: Hromadka T. V. 
Published: August 2005

AbstractThe Complex Variable Boundary Element Method; or CVBEM will be developed with respect to a variable trial function definition over each boundary element. The benefits in using this technique are that the modeling error in matching the prescribed boundary conditions (there is no error in satisfying the Laplace equation) is reduced without the addition of nodal points to the problem discretization. Consequently, the n × n matrix requirements are not increased when using this new approach.
Two‐phase laminar axisymmetric jet flow: Explicit, implicit, and split‐operator approximations
Authors: Aggarwal S. K. , Fix G. J. , Sirignano W. A. 
Published: August 2005

AbstractAn hybrid Eulerian‐Lagrangian numerical scheme is developed for a two‐phase problem and four finite‐difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second‐order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas‐phase equations has been solved by four different numerical methods: a predictor‐corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator‐splitting method. The subsystem of liquid‐phase equations is solved along the droplet trajectories by a second‐order Runge‐Kutta scheme. The computations have been made to predict the hydro‐dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference‐schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.
On operator splitting of the Euler equations consistent with Harten's second‐order accurate TVD scheme
Authors: Cooke C. H. 
Published: August 2005

AbstractOperator splitting is necessary in order to apply Harten's second‐order accurate, total‐variation‐diminishing (TVD) method for shock capturing to higher‐dimensional problems. Here two such splittings of the Euler equations are considered. By analytical means, one splitting is shown to be inconsistent; the other, scheme satisfies a fundamental necessary condition for consistency. Numerical results from a typical blast wave calculation employing this scheme are exhibited.
Nonreflecting upwind boundaries for hyperbolic equations
Authors: Vichnevetsky R. , Pariser E. C. 
Published: August 2005

AbstractIt is commonly, but erroneously, assumed that the best way to treat upwind (closed) boundaries in numerical approximations of hyperbolic equations consists in a literal transcription, letting the numerical value be equal to the prescribed value. This results in a total reflection of spurious solutions that may arrive at the boundary from the computing domain. Those reflected solutions cannot be distinguished from consistent solutions, and they may seriously degrade the overall accuracy. We show that modifications of this treatment of the boundary may result in the absorption of spurious solutions. The effect of the absorbing properties of these boundary schemes is analyzed in Fourier space. We also analyze their numerical stability properties, and their effect on the accuracy of solutions generated in response to a time dependent boundary condition.
Numerical methods for the linear Boltzmann transport equation in slab geometry
Authors: Pao C. V. 
Published: August 2005

AbstractAn iterative method for computing numerical solutions of a finite‐difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite‐difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well‐posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time‐increment the discrete system is numerically stable, independent of the mesh‐size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time‐dependent and the steady‐state solutions.
A semi‐discrete convergent scheme for a quasilinear hyperbolic equation
Authors: Kannan R. , Ortega R. 
Published: August 2005

AbstractWe establish here the convergence (thereby proving the existence) of a semi‐discrete scheme for the quasilinear hyperbolic equation
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where x ∈ Rn, t ∈ [0,T], and ϕ ∈ L∞ (Rn). It is well known that the above problem does not necessarily have global classical solutions and the usual concepts of weak solution. do not lead to a unique solution The existence of a unique solution to the above problem in a suitable sense was established in [3], where a parabolic problem obtained by introducing the term −ϵΔu was studied and then the behavior as ϵ → 0 was discussed. A difference scheme approach to a problem of the above type where ϕi does not depend on x and t and Ψ does not depend on u was also studied in [2]. The aim of this paper is to present a proof for the case when ϕ depends on x, Ψ depends on u, and the technical complications in this case are nontrivial. The discussions in this paper my be considered as continuation of the ideas in the above papers.
Solution of the advection‐dispersion equation with free exit boundary
Authors: Frind Emil O. 
Published: August 2005

AbstractI investigated the exit boundary condition for the advection‐dispersion equation and found that in numerical solutions of this equation, using Galekin finite elements, a free exit boundary condition requiring no a priori information is possible, provided the advective component in the numerical equations is of sufficient magnitude relative to the dispersive component. Since the relationship between these two components is controlled by the spatial discretization through the grid Peclet number, the free exit boundary condition can in fact be applied whenever there is a non‐zero advective component. The numerical solution in a finite domain with free exit boundary, using a correctly proportioned spatial discrezation, behaves like an infinite‐domain solution.
Stability improvement of the Euler forward method for initial value problems
Authors: Kujawski Jerzy 
Published: August 2005

AbstractThe Euler forward method is transformed into a highly stable two‐step explicit algorithm by use of a relaxation paramenter α. The stability of the new explicit family can be very high and linearly proportional to the absolute value of α. An adaptive stability approach with a matrix of parameters α is used that strongly increases the effeciency of the method. The one‐dimensional finite element solutions for linear and nonlinear transient heat conduction problems show the stability, accuracy, and computational efficiency characteristics of the proposed algorithms.
Super‐computer simulation of a multichambered muffler
Authors: Cooke Charlie H. 
Published: August 2005

AbstractThe problem of attenuating muzzle blast noise produced by firing a gun weapon is studied. A multichambered muffler is attached to a M242 25mm cannon. For this weapons configuration, the transitional ballistics problem is solved numerically using the CRAY XMP computer. The second‐order accurate total variation diminishing (TVD) shock capturing scheme of Harten is used to solve the Euler equations of compressible flow. The simulation yields a detailed picture of the flow field, as displayed by pressure and Mach contours. Pressure time histories at selected field locations are recorded for future comparison with experiment. Energy efflux time history, from which predictions of noise attenuation can be made, is also recorded. Noise reduction levels so obtained are seen to compare well with experimental data. From this evidence it appears that the simulation method can become an effective aid in muffler design.
On the accuracy and convergence of conjugate flux approximations
Authors: Rachowicz W. , Oden J. Tinsley 
Published: August 2005

AbstractWe prove that the L2‐projections of derivatives of piecewise bilinear or linear finite element approximations of smooth solutions of elliptic boundary‐value problems on the interior of uniform meshes, converge in L2 and L∞ at a rate faster than that of derivatives of the approximations themselves.
Regularization procedures of mixed finite element approximations of the stokes problem
Authors: Pierre Roger 
Published: August 2005

AbstractWe propose a theoretical framework for the study of regularization of the Stokes problem. This enables us to perform a general error analysis and to apply it to known schemes as well as to a new one pertaining to the use of the P1‐P1 element. Finally we show that in the P1‐case the theory can also be used to get convergence results for elements obtained by addition of bubble functions, without using the usual mixed finite element machinery.
Exact solutions to a finite‐difference model of a nonlinear reaction‐advection equation: Implications for numerical analysis
Authors: Mickens Ronald E. 
Published: August 2005

AbstractThis article gives exact solutions to a finite‐difference model of a nonlinear reaction‐advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.
Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part I: L2‐error estimates
Authors: Goodsell G. , Whiteman J. R. 
Published: August 2005

AbstractSuperconvergence properties in the L2 norm are derived for the recovered gradients of piecewise quadratic finite element approximations on triangular partitions for two‐dimensional elliptic problems and systems, including the case of linear elasticity. The analysis covers problems defined on polygonal domains, where the solutions have low regularity. The effects of numerical integration are treated.
Mixed methods and the marriage between “mixed” finite elements and boundary elements
Authors: Bossavit Alain 
Published: August 2005

AbstractWhen fields that describe a physical situation extend over the whole space, but with complex behavior (nonlinearity, coupling, etc.) only in a bounded region and simple behavior in the rest of space, it may be worthwhile to treat the inner field by a finite elements procedure and the outer field by boundary elements. We address this “marriage” problem in the case of mixed elements. The model problem adopted for this discussion is magnetostatics. A new approach to the question of mixed elements is used, which emphasizes their parenthood with a classical concept of differential geometry, Whitney forms.
Penalty‐combined approaches to the Ritz‐Galerkin and finite element methods for singularity problems of elliptic equations
Authors: Li Zi‐Cai 
Published: August 2005

AbstractPenlty coupling techniques on an interface boundary, artificial or material, are first presented for combining the Ritz–Galerkin and finite element methods. An optimal convergence rate first is proved in the Sobolev norms. Moreover, a significant coupling strategy, L + 1 = O(|ln h|), between these two methods are derived for the Laplace equation with singularities, where L + 1 is the total number of particular solutions used in the Ritz–Galerkin method, and h is the maximal boundary length of quasiuniform elements used in the linear finite element method. Numreical experiments have been carried out for solving the benchmark model: Motz's problem. Both theoretical analysis and numreical experiments clearly display the importance of penalty‐combined methods is solving elliptic equations with singularities.
Optimal convergence rates for the combined methods of different finite element methods
Authors: Li Zi‐Cai 
Published: August 2005

AbstractCoupling techniques are essential to combining different numerical methods together for the purpose of solving an elliptic boundary value problem. By means of nonconforming constraints, the combinations of various Lagrange finite element methods often cause reduced rates of convergence. In this article, we present a method using penalty plus hybrid technique to match different finite element methods such that the optimal convergence rates in the ‖ · ‖h and zero norms of errors of the solution can always be achieved. Also, such a coupling technique will lead to an optimal asymptotic condition number for the associated coefficient matrix. Moreover, this study can easily be extended for combining the finite difference method with the finite element method to also yield the optimal rate of convergence.
Mathematical model of coupled plates meeting at an angle 0 < θ < π and its finite element method
Authors: Lie‐heng Wang 
Published: August 2005

AbstractA mathematical model of coupled plates meeting at an arbitrary angle 0 < θ < π is established. The finite element approximation for the mathematical model is considered, and the optimal error bound is obtained, which shows that the convergence of the approximation is reciprocally proportional to the sine of the angle θ between the two plates. © 1993 John Wiley & Sons, Inc.
On the relationship between nodal schemes and mixed‐hybrid finite elements
Authors: Hennart J. P. , Valle E. Del 
Published: August 2005

AbstractThe relationships between a general family of nodal schemes [J.P. Hennart, SIAM J. Sci. Stat. Comput. 3, 264 (1986)] and primal nonconforming as well as mixed‐hybrid finite element formulations are explored in this paper. It is shown in particular that the nodal schemes proposed are quite natural extensions in rectangular geometry of the Raviart‐Thomas‐Nédélec mixed‐hybrid finite elements. This paper is a revised and extended version with new numerical results of part of a previous research report [J.P. Hennart, Rapports de Recherche INRIA Vol. 386 (1985)]. © 1993 John Wiley & Sons, Inc.
Improved error estimates for mixed finite‐element approximations for nonlinear parabolic equations: The discrete‐time case
Authors: Garcia Sonia M. F. 
Published: August 2005

AbstractThis article treats mixed finite‐element methods for second‐order quasilinear parabolic equations. A fully discrete scheme is presented and L2‐error estimates are established. The convergence of both the function value and the flux is demonstrated. In a previous paper, Part I the author introduced powerful numerical methods for dealing with parabolic problems. The technique used here is an extension of the continuous‐time results for the discrete‐time case. © 1994 John Wiley & Sons, Inc.
Steady incompressible flow around objects in general coordinates with a multigrid solution method
Authors: Oosterlee C. W. , Wesseling P. 
Published: August 2005

AbstractSteady incompressible flow around objects in general coordinates is investigated. First, an overview of the popular approaches to discretize incompressible flow problems in general coordinates is given. It has been chosen to solve the equations on a staggered grid with contravariant flux unknowns and pressure as primitive variables. A solution method multigrid is used, with a line smoother able to deal with stretched cells. For flow problems around objects solved with a single block discretization periodic boundary, conditions are prescribed and adaptations for the discretization and the multigrid method are given. Steady flow around a circular cylinder and around an ellipse are presented. © 1994 John Wiley & Sons, Inc.
Analysis of the Kleiser‐Schumann method for a family of mixed finite elements for the Stokes equations
Authors: Chinosi Claudia , Comodi Maria I. 
Published: August 2005

AbstractWe study the Stokes problem by using a mixed finite element method introduced by Stenberg [Numer. Math. 56, 827 (1990)] and adapt it to the splitting introduced by Canuto and Sacchi Landriani [Numer. Math. 50, 217 (1986)] and Chinosi and Comodi [Numer. Methods Partial Different. Equations 7, 363 (1991)], by using the incompressibility condition to implicitly define a boundary condition for the pressure. © 1994 John Wiley & Sons, Inc.
Numerical solutions of the Dirichlet problem via a density theorem
Authors: Whitley Robert , Hromadka T. V. 
Published: August 2005

AbstractUnder mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for Lp boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.
Convergence of the lines method for first‐order partial differential‐functional equations
Authors: Zubik‐Kowal Barbara 
Published: August 2005

AbstractInitial‐boundary value problems for nonlinear differential‐functional equations are considered. A general class of lines method is investigated. The Perron‐type estimation for the right‐hand side of the equation with respect to the functional argument is assumed. The proof of the convergence is based on a comparison theorem for differential‐difference inequalities. A numerical example is given. © 1994 John Wiley & Sons, Inc.
Optimal‐order convergence rates for Eulerian‐Lagrangian localized adjoint methods for reactive transport and contamination in groundwater
Authors: Wang Hong , Ewing R. E. 
Published: August 2005

AbstractEulerian–Langrangian localized adjoint methods (ELLAM) were developed to solve convection–diffusion–reaction equations governing contaminant transport in groundwater flowing through a porous medium, subject to various combinations of boundary conditions. In this article, we prove optimal‐order error estimates and some superconvergence results for the ELLAM schemes. In contrast to many existing estimates for a variety of numerical methods, which often contain the temporal derivatives of the exact solution, our error estimates contain the total derivatives of the exact solution but do not involve any temporal derivatives of the exact solution. © 1995 John Wiley & Sons, Inc.
A primal hybrid finite element method for a strongly nonlinear second‐order elliptic problem
Authors: Park E.‐J. 
Published: August 2005

AbstractThe Neumann problem for a strongly nonlinear second‐order elliptic equation in divergence form is approximated by primal hybrid finite element methods defined by Raviart and Thomas. Existence and uniqueness of the approximation are proved, and optimal order error estimates are established in various norms. © 1995 John Wiley & Sons, Inc.
Interpolation for solutions of the Helmholtz equation
Authors: González–Casanova Pedro , Wolf Kurt Bernardo 
Published: August 2005

AbstractWe study the interpolation problem for solutions of the two‐dimensional Helmholtz equation, which are sampled along a line. The data are the function values and the normal derivatives at a discrete set of point sensors. A wave transform is used, analogous to the common Fourier transform. The inverse wave transform defines the Hilbert space for oscillatory Helmholtz solutions. We thereby introduce an interpolant that has some advantages over the usual sinc x in the Whittaker–Shannon sampling in one dimension; in particular, coefficients of the two‐dimensional solution are invariant under translations and rotations of the sampling line. The analysis is relevant for the optical sampling problem by sensors on a screen. © 1995 John Wiley & Sons, Inc.
The stability of finite element methods
Authors: Verfürth Rüdiger 
Published: August 2005

AbstractWe give an abstract stability result for finite element methods. It provides a general guideline for establishing the stability of finite element methods for a given norm once it is established with respect to another norm. The abstract result yields stability estimates of natural norms for finite element methods, which have recently been developed to stabilize effects like dominant convection, incompressibility constraints, or boundary conditions. These stability results simplify the error analysis of the discretizations and are helpful fin the convergence analysis of multigrid methods for the solution of the corresponding discrete problems. © 1995 John Wiley & Sons, Inc.
Least‐squares finite element approximation of Fisher's reaction–diffusion equation
Authors: Carey G. F. , Shen Yun 
Published: August 2005

AbstractFisher's equation is a classical diffusion–reaction type of problem describing diffusion and nonlinear reproduction of a species. In the present study we develop a least‐squares finite element formulation of Fisher's equation and carry out supporting numerical studies. Of particular interest are questions associated with the approximation of progressive wave solutions with minimum speed and the viability of the least‐squares approach for this class of problem. © 1995 John Wiley & Sons, Inc.