bluuebunny/crossref_metadata_2025_split huggingface dataset:
Asymptotic stability of the numerical solution for an integrodifferential reaction‐diffusion system
Authors: Galeone L. , Mastroserio C. , Montrone M. 
Published: August 2005

AbstractThis paper presents the study of the numerical solution of a reaction‐diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M‐matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.
Preconditioned iterative algorithms for large sparse unsymmetric problems
Authors: Pini G. , Zilli G. 
Published: August 2005

AbstractA numerical study of the efficiency of the generalized conjugate residual methods (GCR) is performed using three different preconditioners all based upon an incomplete LU factorization. The GCR behavior is evaluated in connection with the solution of large, sparse unsymmetric systems of equations, arising from the finite element integration of the diffusion‐convection equation for 2‐dimensional (2‐D) and 3‐D problems with different Peclet and Courant numbers. The order of the test matrices ranges from 450 to 1700. Results from a set of numerical experiments are presented and comparisons with preconditioned GCR methods and with direct method are carried out.
Approximating solutions to linear and nonlinear differential equations by the method of maximum entropy
Authors: Baker‐Jarvis James , Schultz Dwight , Alamendine Jihad 
Published: August 2005

AbstractA technique to approximate the solution to linear and nonlinear boundary value problems is developed and numerical examples are presented. The technique is based on the method of maximum entropy with moments of the differential equation used as constraints. The method is very general and has the advantage that additional information can be fed into the solution, such as the function's domain or the positivity or negativity of the solution. The technique should find applications in approximating solutions to equations which may or may not contain noise and as an alternative to finite difference and Fourier series solutions and may have applications to large scale simulations.
Mixed‐hybrid finite elements and streamline computation for the potential flow problem
Authors: Kaasschieter E. F. , Huijben A. J. M. 
Published: August 2005

AbstractAn important class of problems in mathematical physics involves equations of the form −∇ · (A∇ϕ) = f. In a variety of problems it is desirable to obtain an accurate approximation of the flow quantity u = −A∇ϕ. Such an accurate approximation can be determined by the mixed finite element method. In this article the lowest‐order mixed method is discussed in detail. The mixed finite element method results in a large system of linear equations with an indefinite coefficient matrix. This drawback can be circumvented by the hybridization technique, which leads to a symmetric positive‐definite system. This system can be solved efficiently by the preconditioned conjugate gradient method. After approximating u by the lowest‐order mixed finite element method, streamlines and residence times can be determined easily and accurately by computations at the element level.
Probabilistic viscosity algorithm for the scalar conservation law
Authors: Korzeniowski Andrzej 
Published: August 2005

AbstractWe derive an algorithm for solving the initial value problem for ut = ½σ2uxx + f(u)ux. The approach is based on the representation of the solution to the above equation in the form of the functional of Brownian motion. For small σ we get the approximation for ut = f(u)ux. A comparison with the random choice method is illustrated by the numerical example.
Box‐spline–based approach to the formulation of numerical methods for partial differential equations
Authors: Diamond Harvey , Raphael Louise Arakelian , Williams Daniel A. 
Published: August 2005

AbstractA multivariate box spline framework for the formulation of numerical methods for partial differential equations has been constructed. In particular, a fourth‐order Galerkin method and a second‐order collocation method were derived and applied to a test problem (classical Poisson equation on a square). The examples indicate that accuracy compares favorably with standard methods and the success of iterative schemes suggests an underlying stabilizing effect.
Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions
Authors: Penzel F. 
Published: August 2005

AbstractWe present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.
Unified symmetric finite element and boundary integral variational coupling methods for linear fluid–structure interaction
Authors: Zeng Xiaogang , Bielak Jacobo , Maccamy Richard C. 
Published: August 2005

AbstractThis article concerns the development of energy‐based variational formulations and their corresponding finite element–boundary element Rayleigh–Ritz approximations for solving the time‐harmonic vibration and scattering problem of an inhomogeneous penetrable fluid or solid object immersed in a compressible, inviscid, homogeneous fluid. The resulting coupled finite element and boundary integral methods (FEM‐BEM) have the following attractive features: (1) Separate direct and complementary variational principles lead naturally to several alternative structure variable and fluid variable methodologies. (2) The solution in the exterior region is represented by a combined single‐ and double‐layer potential which ensures the validity of the methods for all wave numbers; even though this representation introduces hypersingular integrals, for actual computations the hypersingular operator may be rewritten in terms of single‐layer potentials, which can be integrated by standard techniques. (3) Since the discretized equations for the interior region and for the boundary are derived from the first variation of bilinear functionals the resulting algebraic systems of equations are always symmetric. In addition, the transition conditions across the interface are natural. This allows one to approximate the solutions within the interior and exterior regions independently, without imposing any boundary constraints.
Defect correction and Galerkin's method for second‐order boundary value problems
Authors: Chibi Ahmed‐Salah , Moore Gerald 
Published: August 2005

AbstractThe defect correction technique, based on the Galerkin finite element method, is analyzed as a procedure to obtain highly accurate numerical solutions to second‐order elliptic boundary value problems. The basic solutions, defined over a rectangular region Ω, are computed using continuous piecewise bilinear polynomials on rectangles. These solutions are O(h2) accurate globally in the second‐order discrete Sobolev norm. Corrections to these basic solutions are obtained using higher‐order piecewise polynomials (Lagrange polynomials or splines) to form defects. An O(h2) improvement is gained on the first correction. The lack of regularity of the discrete problems (beyond the second‐order Sobolev norm) makes it impossible to retain this order of improvement, but for problems satisfying certain periodicity conditions, straightforward arbitrary accuracy is obtained, since these problems possess high‐order regularity. © 1992 John Wiley & Sons, Inc.
On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal Padé approximations
Authors: Khaliq A. Q. M. , Twizell E. H. , Voss D. A. 
Published: August 2005

AbstractIt is well known that methods for solving semidiscretized parabolic partial differential equations based on the second‐order diagonal [1/1] Padé approximation (the Crank–Nicolson or trapezoidal method) can produce poor numerical results when a time discretization is imposed with steps that are “too large” relative to the spatial discretization. A monotonicity property is established for all diagonal Padé approximants from which it is shown that corresponding higher‐order methods suffer a similar time step restriction as the [1/1] Padé. Next, various high‐order methods based on subdiagonal Padé approximations are presented which, through a partial fraction expansion, are no more complicated to implement than the first‐order implicit Euler method based on the [0/1] Padé approximation; moreover, the resulting algorithms are free of a time step restriction intrinsic to those based on diagonal Padé approximations. Numerical results confirm this when various test problems from the literature are implemented on a Multiple Instruction Multiple Data (MIMD) machine such as an Alliant FX/8. © 1993 John Wiley & Sons, Inc.
Some results on the stability and dynamics of finite difference approximations to nonlinear partial differential equations
Authors: Choudhury S. Roy 
Published: August 2005

AbstractA miscellany of results on the nonlinear instability and dynamics of finite difference discretizations of the Burgers and Kortweg de Vries equations is obtained using a variety of phase‐plane, functional analytic, and regularity methods. For the semidiscrete (space‐discrete, time‐continuous) schemes, large‐wave‐numer instabilities occurring in special exact solutions are investigated, and parameter values for which the semidiscrete scheme is monotone are considered. For fully discrete schemes (space and time discrete), large‐wave‐number instabilities introduced by various time‐stepping schemes such as forward Euler, leapfrog, and Runge–Kutta schemes are analyzed. Also, a time step restriction for the monotonicity of the forward‐Euler time‐stepping scheme, and regularity of a 4‐stage monotone/conservative Runge–Kutta time stepping are investigated. The techniques used here may be employed, in conjunction with bifurcation‐theoretic and weakly nonlinear analyses, to analyze the stability of numerical schemes for other nonlinear partial differential equations of both dissipative and dispersive varieties. © 1993 John Wiley & Sons, Inc.
An organization of sparse gauss elimination for solving partial differntial equations on distributed memory machines
Authors: Mu Mo , Rice John R. 
Published: August 2005

AbstractA computational arrangement of Gauss elimination is presented for solving sparse, nonsymmetric linear systems arising from partial differential equation problems. It is particularly targeted for use on distributed memory message passing multiprocessor computers and it is presented and analyzed in this context. The objective of the algorithm is to exploit the sparsity (i.e., reducing computation, communication, and memory requirements) and to optimize the data structure manipulation overhead. The algorithm is based on the nested dissection approach, which starts with a large set of very sparse, completely independent subsystems and progresses in stages to a single, nearly dense system at the last stage. The computational efforts of each stage are roughly equal (almost exactly equal for model problems), yet the data structures appropriate for the first and last stages are quite different. Thus we use different types of data structures and algorithm components at different stages of the solution. The new organization is a combination of previous techniques including nested dissection, implicit block factorization, domain decomposition, fan‐in, fan‐out, up‐looking, down‐looking, and dynamic data structures. © 1993 John Wiley & Sons, Inc.
Numerical solution of partial differential equations on curved domains by collocation
Authors: Van Blerk J. J. , Botha J. F. 
Published: August 2005

AbstractThe application of the finite element collocation method to two‐ and three‐dimensional partial differential equations is hampered by the fact that its accuracy depends largely on the position of the collocation points. The method has, in the past, thus mainly been applied to two‐dimensional differential equations defined on rectangular domains. In this case the method yields an optimal global convergence rate, when the Gauss‐Legendre quadrature points are used as collocation points. It is shown in this paper that the same accuracy can also be achieved in the case of differential equations defined on nonrectangular domains. The only prerequisite for this is to solve the differential equation on a rectangular domain, mapped onto the nonrectangular domain of the differential equation by a bilinear blended map. © 1993 John Wiley & Sons, Inc.
Studies of an interface relaxation domain decomposition technique using finite elements on a parallel computer
Authors: Wang K. P. , Bruch J. C. 
Published: August 2005

AbstractStudies are presented for an interface relaxation domain decomposition technique using finite elements on an iPSC/2 D5 Hypercube Concurrent computer. The general type of problem to be solved is one governed by a partial differential equation. The application of the approach, however, will be extended to a free boundary value problem by appropriate modification of the numerical scheme. Using the domain decomposition technique, the computation domain is subdivided into several subdomains. In addition, on the interfaces between two adjacent subdomains are imposed a continuity condition on one side and an equilibrium condition on the other side. Successive overrelaxation iterative processes are then carried out in all subdomains with a relaxation process imposed on the interfaces. With this domain decomposition technique, the problem can be solved parallelly until convergence is reached both in the interiors and on the interfaces of all subdomains. Moreover, the formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee load balancing. Furthermore, it is shown, through numerical experiments performed on an example problem of free surface seepage through a porous dam, how the values of the relaxation parameters, the choice of imposed boundary conditions, and the number of subdomains (i.e., the number of processors used) affect the solution convergence in this parallel computing environment. © 1993 John Wiley & Sons, Inc.
Numerical approximation of bounded solutions for semilinear elliptic equations in an unbounded cylindrical domain
Authors: Fuming Ma 
Published: August 2005

AbstractWe consider the approximation of bounded solutions for a class of semilinear elliptic equations in an unbounded cylindrical domain. On certain conditions, the existence of approximate solutions for the semidiscretization of the problem is proved. Error estimates, applications, and a numerical example are also given. © 1993 John Wiley & Sons, Inc.
Error analysis of the finite‐strip method for parabolic equations
Authors: Smith Stanley S. , Allen Myron B. 
Published: August 2005

AbstractThe finite‐strip method (FSM) is a hybrid technique which combines spectral and finite‐element methods. Finite‐element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted‐residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time‐dependent, parabolic partial differential equations indicates the numerical solution error is O(h2 + M−r). M represents the Fourier truncation mode number and h represents the finite‐element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite‐layer method, where a two‐dimensional trigonometric basis is used, the numerical solution error is O(h2 + M−r + N−q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley & Sons, Inc.
Superconvergence of recovered gradients of discrete time/piecewise linear Galerkin approximations for linear and nonlinear parabolic problems
Authors: Wheeler Mary F. , Whiteman J. R. 
Published: August 2005

AbstractSuperconvergent error estimates in l2(H1) and l∞(H1) norms are derived for recovered gradients of finite difference in time/piecewise linear Galerkin approximations in space for linear and quasinonlinear parabolic problems in two space dimensions. The analysis extends previous results for elliptic problems to the parabolic context, and covers problems in regions with nonsmooth boundaries under certain assumptions on the regularity of the solutions. © 1994 John Wiley & Sons, Inc.
Crank‐Nicolson method for the numerical solution of models of excitability
Authors: López‐Marcos J. C. 
Published: August 2005

AbstractWe analyze a Crank‐Nicolson scheme for a family of nonlinear parabolic partial differential equations. These equations cover a wide class of models of excitability, in particular the Hodgkin Huxley equations. To do the analysis, we have in mind the general discretization framework introduced by López‐Marcos and Sanz‐Serna [in Numerical Treatment of Differential Equations, K. Strehemel, Ed., Teubner‐Texte zur Mathematik, Leipzig, 1988, p. 216]. We study consistency, stability and convergence properties of the scheme. We use a technique of modified functions, introduced by Strang [Numer. Math. 6, 37 (1964)], in the study of consistency. Stability is derived by means of the energy method. Finally we obtain existence and convergence of numerical approximations by means of a result due to Stetter (Analysis of Discretization Methods for Ordinary Differential Equations. Springer‐Verlag, Berlin, 1973). We show that the method has optimal order of accuracy in the discrete H1 norm. © 1994 John Wiley & Sons, Inc.
Viscous stress relaxation, characteristics, and numerical boundary conditions
Authors: Trapp John A. , Ransom Victor H. 
Published: August 2005

AbstractThe anomalous infinite propagation speeds in the classical parabolic flow equations are removed by the inclusion of a small amount of fluid elasticity or viscous stress relaxation. The inclusion of such effects results in a hyperbolic system of equations with a complete set of characteristic equations. The directional characteristic equations are used to give insights into the appropriate boundary molecules to be used in finite difference numerical schemes. © 1994 John Wiley & Sons, Inc.
A fourth‐order finite difference scheme for two‐dimensional nonlinear elliptic partial differential equations
Authors: Saldanha Godfrey , Ananthakrishnaiah U. 
Published: August 2005

AbstractA finite difference scheme for the two‐dimensional, second‐order, nonlinear elliptic equation is developed. The difference scheme is derived using the local solution of the differential equation. A 13‐point stencil on a uniform mesh of size h is used to derive the finite difference scheme, which has a truncation error of order h4. Well‐known iterative methods can be employed to solve the resulting system of equations. Numerical results are presented to demonstrate the fourth‐order convergence of the scheme. © 1995 John Wiley & Sons, Inc.
Efficient computation of a mathematical model for the damping of pressure waves in tubes of circular form
Authors: Kroller M. 
Published: August 2005

AbstractA computational method for a system of partial integrodifferential equations of Volterra type is developed. These equations describe the damping of pressure waves in tubes of circular form, taking into account the nonstationary increase of friction. The analytical solution of the model equations is given. A rapidly convergent expansion as well as sharp error estimates for the approximate solutions are obtained. Finally an efficient algorithm using recursion formulas is presented and the algorithm is compared numerically with the method of characteristics. © 1995 John Wiley & Sons, Inc.
Finite element approximation of optimal control problems for the von Kármán equations
Authors: Hou L. Steven , Turner James C. 
Published: August 2005

AbstractThis article is concerned with optimal control problems for the von Kármán equations with distributed controls. We first show that optimal solution exist. We then show that Lagrange multipliers may be used to enforce the consstraints and derive an optimality system from which optimal states and controls may be deduced. Finally, we define finite element approximations of solutions for the optimality system and derive error estimate for the approximations. © 1995 John Wiley & Sons, Inc.
Finite difference reaction diffusion equations with nonlinear boundary conditions
Authors: Pao C. V. 
Published: August 2005

AbstractThis article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time‐dependent and steady‐state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence‐comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time‐dependent solution in relation to the steady‐state solutions. Application is given to a heat‐conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady‐state solutions, and determines the dynamic behavior of the time‐dependent solution. Numerical results for the heat‐conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.
Fractional step algorithm for solving a multi‐dimensional diffusion‐migration equation
Authors: Kwok Yue‐Kuen , Wu Charles C. K. 
Published: August 2005

AbstractThe earlier fractional step algorithm for solving the diffusion–migration equation in electrochemistry is extended to a multi‐dimensional multi‐species system with second‐order spatial accuracy. For each time‐step increment, the algorithm consists of three stages: (i) diffusion, (ii) satisfaction of the electroneutrality constraint, and (iii) migration. Each stage accounts for one individual physical process. Exact analytical solutions are derived for a two‐species system and comparisons between exact and numerical results are made. Numerical results are also obtained for a two‐dimensional three‐species electrochemical model. © 1995 John Wiley & Sons, Inc.