bluuebunny/crossref_metadata_2025_split huggingface dataset:
A regularization‐stabilization technique for nonlinear conservation equation computations
Authors: Carey G. F. , Jiang B. N. , Showalter R. 
Published: August 2005

AbstractA regularization procedure for nonlinear conservation equations is introduced and demonstrated to have a stabilizing effect on the numerical solution of the associated approximate problem. Representative results for a least‐squares finite‐element method are given, and the numerical performance of the stabilization procedure explored. The effect of the regularization term is similar to a local numerical dissipation dependent on the numerical itegration time step.
Six combinations of the Ritz‐Galerkin and finite element methods for elliptic boundary value problems
Authors: Li Z. C. , Bui T. D. 
Published: August 2005

AbstractOverall comparisons are made for six efficient combinations of the Ritz‐Galerkin and finite element methods for solving elliptic boundary value problems with singularities or interfaces. The comparisons are done by using theoretical analysis and numerical experiments. Significant relations among the six combinations are also found. A survey of the six combinations and their coupling strategies are given. These combinations are important not only for matching the Ritz‐Galerkin method and the finite element method but also for matching other numerical methods such as the Ritz‐Galerkin method and the finite difference method.
Numerical solution of the Fokker Planck equation using moving finite elements
Authors: Harrison Gray W. 
Published: August 2005

AbstractNumerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion. Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large. A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process. A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation. Solutions of two test equations are presented to illustrate the effectiveness of the method.
On a zero equation model of turbulence
Authors: Turner James C. 
Published: August 2005

AbstractIn this paper we study a widely used zero equation model of turbulence. The governing equations are derived by applying to the incompressible Navier‐Stokes equations the Reynolds time averaging procedure. We achieve closure by employing the eddy viscosity concept. Using the Implicit Function Theorem we obtain an existence and uniquencess result. We also discuss the existence of nonsingular solutions. Finally, we present an algorithm for solving the modeled equations.
A new numerical approach for the advective‐diffusive transport equation
Authors: Celia Michael A. , Herrera Ismael , Bouloutas Efthimios , Kindred J. Scott 
Published: August 2005

AbstractA new numerical solution procedure is presented for the one‐dimensional, transient advective‐diffusive transport equation. The new method applies Herrera's algebraic theory of numerical methods to the spatial derivatives to produce a semi‐discrete approximation. The semi‐discrete system is then solved by standard time marching algorithms. The algebraic theory, which involves careful choice of test functions in a weak form statement of the problem, leads to a numerical approximation that inherently accommodates different degrees of advection domination. Algorithms are presented that provide either nodal values of the unknown function or nodal values of both the function and its spatial derivative. Numerical solution of several test problems demonstrates the behavior of the method.
An approach combining the Ritz‐Galerkin and finite difference methods
Authors: Li Zi‐Cai 
Published: August 2005

AbstractThe nonconforming combination of Ritz‐Galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities. The Ritz‐Galerkin method is used in the subdomains including singularities, the finite difference method is used in the rest of the solution domain. Moreover, on the common boundary of two regions where two different methods are used, the continuity conditions are constrained only on the nodes of difference grids. Theoretical analysis and numerical experiments have shown that average errors of numerical solutions and their generalized derivatives can reach the convergence rate O(h2‐δ), where h is the mesh spacing of uniform difference grids, and δ is an arbitrarily small, positive number. This convergence rate is better than O(h), obtained by the nonconforming combination of the Ritz‐Galerkin and finite element methods.
Recovery of an unknown specific heat by means of overposed data
Authors: Pilant Michael , Rundell William 
Published: August 2005

AbstractIn this paper we consider a class of inverse problems in which an unknown function, c(.), is to be determined from a parabolic initial‐value problem, with overposed Dirichlet data along a portion of the boundary. A mapping between the overposed data and the unknown coefficient is obtained in the form of a singular integral equation. This is solved by iteration, and the resulting fixed point is shown to be the solution of the inverse problem. Sufficient conditions for convergence of this method, as well as an extension to the case of an unknown thermal conductivity, are given.
Numerical approaches for computation of fronts
Authors: Sheintuch Moshe 
Published: August 2005

AbstractMoving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady‐state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary‐value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low‐order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three‐element structure where the edge‐elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.
Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity
Authors: Goodsell G. , Whiteman J. R. 
Published: August 2005

AbstractIn this paper we consider piecewise linear finite element approximations to the problem of planar linear elasticity, and present some new estimates for the pointwise (L∞) superconvergence of a recovered gradient function to the gradient of the true solution. This extends to linear elasticity the previous work of the present and other authors on L∞ results for Poisson problems, and at the same time, to the L∞ norm the previous L2 results of the authors for linear elasticity.
An analysis of the finite element method for natural convection problems
Authors: Boland J. , Layton W. 
Published: August 2005

AbstractWe derive stability properties and error estimates for the finite element method when used to approximate heat flow in a fluid enclosed by a solid medium. The coupled Navier Stokes system involves the Boussinesq equations in the fluid‐filled cavity linked through an interface with heat conduction in the solid enclosing the fluid. As we assume no extra regularity then can be shown to hold under mild restriction on the data (at least over a small time interval in R3), we focus primarily on low order finite element spaces.
Path integral numerical methods for evolution equations
Authors: Miranker Willard L. , Winkler Andrew 
Published: August 2005

AbstractA universal method for representing solutions of the initial value problem ut = −2πiDu is presented for a large class of pseudo‐differential operators. The representations take the form of a path integral‐like operator. Three numerical approaches for evaluating these operators are treated. Practical interest in such universal methods depends on massively parallel computing systems.
Numerical solutions of some parabolic inverse problems
Authors: Cannon John R. , Yin Hong‐Ming 
Published: August 2005

AbstractIn this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we introduce a variational form by defining a new function. Both continuous and discrete Galerkin procedures are illustrated in this paper. The error estimates are also derived.
Generalized alternating‐direction collocation methods for parabolic equations. I. Spatially varying coefficients
Authors: Celia Michael A. , Pinder George F. 
Published: August 2005

AbstractThe alternating‐direction collocation (ADC) method can be formulated for general parabolic partial differential equations. This is done using a piecewise cubic Hermite trial space defined on a rectangular discretization. As in all alternating‐direction methods, the ADC algorithm produces errors that are additional to the standard discretization errors of multi‐dimensional collocation. These errors increase when the coefficients of the governing equation are spatially variable. Analysis of the additional errors leads to several correction schemes. Numerical results indicate that a variant on the Laplace‐modification procedure is an attractive choice as an improved ADC algorithm.
Generalized alternating‐direction collocation methods for parabolic equations. II. Transport equations with application to seawater intrusion problems
Authors: Celia Michael A. , Pinder George F. 
Published: August 2005

AbstractThe alternating‐direction collocation (ADC) method combines the attractive computational features of a collocation spatial approximation and an alternating‐direction time marching algorithm. The result is a very efficient solution procedure for parabolic partial differential equations. To date, the methodology has been formulated and demonstrated for second‐order parabolic equations with insignificant first‐order derivatives. However, when solving transport equations, significant first‐order advection components are likely to be present. Therefore, in this paper, the ADC method is formulated and analyzed for the transport equation. The presence of first‐order spatial derivatives leads to restrictions that are not present when only second‐order derivatives appear in the governing equation. However, the method still appears to be applicable to a wide variety of transport systems. A formulation of the ADC algorithm for the nonlinear system of equations that describes density‐dependent fluid flow and solute transport in porous media demonstrates this point. An example of seawater intrusion into coastal aquifers is solved to illustrate the applicability of the method.An alternating‐direction collocation solution algorithm has been developed for the general transport equation. The procedure is analogous to that for the model parabolic equations considered by Celia and Pinder [2]. However, the presence of first‐order spatial derivatives requires special attention in the ADC formulation and application. With proper implementation, the ADC procedure effectively combines the efficient equation formulation inherent in the collocation method with the efficient equation solving characteristics of alternating‐direction time marching algorithms. To demonstrate the viability of the method for problems with complex velocity fields, the procedure was applied to the problem of density‐dependent flow and contaminant transport in groundwaters. A standard example of seawater intrusion into coastal aquifers was solved to illustrate the applicability of the method and to demonstrate its potential use in practical problems.
Generalized alternating‐direction collocation methods for parabolic equations. III. Nonrectangular domains
Authors: Celia Michael A. , Pinder George F. 
Published: August 2005

AbstractThe alternating‐direction collocation (ADC) method is an efficient numerical approximation technique for the solution of parabolic partial differential equations. However, to date the ADC method has only been developed for rectangular discretizations. With judicious combination of isoparametric coordinate transformations and an extended ADC approach, the ADC method can be formulated on general nonrectangular domains. This extends the applicability of the ADC method by allowing it to be employed on domains of more general geometry.
The effect of numerical integration in finite element methods for nonlinear parabolic equations
Authors: Chou So‐Hsiang , Li Qian 
Published: August 2005

AbstractWe consider the effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations and give some sufficient conditions to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal error estimates for the solution and its time derivative in the spaces L∞(H1) and L∞(L2) are established. Moreover, maximum norm error estimates are demonstrated.
Characteristic adaptive subdomain methods for reservoir flow problems
Authors: Dahle H. K. , Espedal M. S. , Ewing R. E. , Sævereid O. 
Published: August 2005

AbstractThe simulation of multiphase and multicomponent fluid flows often requires the solution of large, coupled systems of nonlinear partial differential equations of convection‐advection type. The equations are usually advection dominated with small but important local diffusive effects. An operator‐splitting technique is used to address these different phenomena. A modified method of characteristics is used to try to eliminate the nonsymmetry in the operators due to advection. Although this method does not completely symmetrize the problem with nonlinear, nonconvex flux functions, the remaining operator is almost symmetric and can be stabilized via Petrov‐Galerkin techniques with optimal or near optimal test functions. Effective localized test functions are descrbed. A coarse grid is defined to treat the slow variation of the fluid velocity away from fronts. A substructuring of this grid gives a proper frontal resolution via local patch grid refinement. Preconditioning techniques are presented to efficiently solve the resulting composite‐grid problem. Error estimates for the total approximation procedure are presented. Finally, certain two‐dimensional computations are described.
Influence matrix technique for the numerical spectral simulation of viscous incompressible flows
Authors: Phillips Timothy N. , Soliman Ibrahim M. 
Published: August 2005

AbstractA spectral algorithm based on the influence matrix technique is desclibed for solving numerically the flow of incompressible viscous fluids. The algorithmic development is for both Newtonian and non‐Newtonian flows. To investigate the performance of the method several test problems are solved. Accurate results are obtained with relatively few degrees of freedom.
Prismatic versus tetrahedral elements in three‐dimensional finite element analyses of subsurface systems
Authors: Pini G. 
Published: August 2005

AbstractA mesh of prismatic or tetrahedral elements automatically generated from an initial triangular grid is used to integrate 3‐D flow equation in space. Many numerical comparisons between these two models have been performed. The results show that integration with tetrahedrons is as accurate as integration with prisms but much more efficient. The CPU time of solution with prismatic elements is about three times greater than that required employing tetrahedral elements.
Fourth‐order difference methods for the system of 2D nonlinear elliptic partial differential equations
Authors: Jain M. K. , Jain R. K. , Mohanty R. K. 
Published: August 2005

AbstractWe present the fourth‐order finite difference methods for the system of 2D nonlinear elliptic equations using 9‐grid points on a square region R subject to Dirichlet boundary conditions. The method has been tested on viscous, incompressible 2D Navier‐Stokes equations. The numerical results show that the proposed methods produce accurate and oscillation‐free solutions for large Reynolds numbers.
Natural and postprocessed superconvergence in semilinear problems
Authors: Chow S.‐S. , Carey G. F. , Lazarov R. D. 
Published: August 2005

AbstractSuperconvergence error estimates are established for a class of semilinear problems defined by a linear elliptic operator with a nonlinear forcing term. The analysis is for rectangular biquadratic elements, and we prove superconvergence of the derivative components along associated lines through the Gauss points. Derivative postprocessing formula and formulas for integrals are also considered and similar superconvergence estimates proven.
Convergence of the contour dynamics method
Authors: Soler Juan 
Published: August 2005

AbstractIn this paper the contour dynamics method (CDM) is formulated and applied to the study of a two‐dimensional incompressible and inviscid fluid flow. A result concerning the convergence of this method is proved. The foundation of the CDM is based on the study of the dynamic interaction between contours enclosing vortex regions. The idea of the analysis is to show that the CDM can be considered as a vortex filament method. Hence, the convergence for the contour dynamics method is proved by using some results in the theory of the vortex method.
Discontinuous solutions to hyperbolic systems under operator splitting
Authors: Roe P. L. 
Published: August 2005

AbstractTwo‐dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time‐steps exactly solves the component one‐dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high‐resolution computing schemes are discussed.
Fourth‐order finite difference method for 2D parabolic partial differential equations with nonlinear first‐derivative terms
Authors: Jain M. K. , Jain R. K. , Mohanty R. K. 
Published: August 2005

AbstractWe attempt to obtain a two‐level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion‐convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.
Simple nonconforming quadrilateral Stokes element
Authors: Rannacher R. , Turek S. 
Published: August 2005

AbstractA simple nonconforming quadrilateral Stokes element based on “rotated” multi‐linear shape functions is analyzed. On strongly nonuniform meshes the usual parametric version of this element suffers from a lack of consistency, while its nonparametric counterpart turns out to be convergent with optimal orders. This theoretical result is confirmed by numerical tests.
Stability analysis of discrete generalized Stokes problems
Authors: Fortin M. , Pierre R. 
Published: August 2005

AbstractIn the mixed finite element approximation of Stokes‐like problems, stability considerations normally restrict the choice of the finite element subspaces to families satisfying the so‐called inf‐sup condition. However, in the case of the generalized Stokes problems that one encounters when solving either the nonstationary compressible or incompressible Navier–Stokes equations by means of some operator splitting, the contribution of the time derivative is adding a term in the continuity equations which, at least theoretically, leads to a problem that is always stable. On the other hand, the numerical solution of the discrete problems built with approximations not satisfying the inf‐sup condition are generally plagued with oscillations that look pretty much like the checkerboard phenomenon occurring in the classical context. To clarify this apparent contradiction, we undertook a theoretical study of the condition number of the linear problem involving the sensitive variable, which for such a saddle point problem is the dual problem. In the course of this study, we have exhibited the relationship between that condition number and the inf‐sup constant, thus providing a different point of view on the results of Brezzi and Babuska. Finally, we conducted a parametric study of the behavior of that condition number with respect to the mesh size, the viscosity, and the time step. That study showed that, even in this context, the use of “stable element” was adding to the quality of the approximation by reducing the condition number to an acceptable level.
Domain‐decomposition approach to local grid refinement in finite element collocation
Authors: Curran Mark C. , Allen Myron B. 
Published: August 2005

AbstractAdvection‐dominated flows occur widely in the transport of groundwater contaminants, the movements of fluids in enhanced oil recovery projects, and many other contexts. In numerical models of such flows, adaptive local grid refinement is a conceptually attractive approach for resolving the sharp fronts or layers that tend to characterize the solutions. However, this approach can be difficult to implement in practice. A domain decomposition method developed by Bramble, Ewing, Pasciak, and Schatz, known as the BEPS method, overcomes many of the difficulties. We demonstrate the applicability of BEPS ideas to finite element collocation on trial spaces of piecewise Hermite cubics. The resulting scheme allows one to refine selected parts of a spatial grid without destroying algebraic efficiencies associated with the original coarse grid. We apply the method to steady‐state problems with boundary and interior layers and a time‐dependent advection‐diffusion problem.
The primal mixed finite element method and the LBB condition
Authors: Robey Thomas H. 
Published: August 2005

AbstractSpurious or kinematic modes have posed a major obstacle to the implementation of the mixed finite element method. This research shows that spurious modes resulting from the approximation spaces not satisfying the LBB condition do not prevent a well posed problem. When the LBB condition is not satisfied, the resulting matrix equations are singular. A direct solution method is presented for the efficient solution of the possibly singular equations. Orthogonal flux basis functions are introduced to simplify the problem. Then the solution procedure is based on nested domain decomposition. This solution procedure is shown to be competitive with direct solution methods for the displacement finite element method. Examples are included to demonstrate various aspects of the LBB condition and the solution procedure.