bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the dynamics of a discrete reaction‐diffusion system
Authors: Azmy Y. Y. , Protopopescu V. 
Published: August 2005

AbstractWe investigate various aspects of the dynamics of a discrete reaction‐diffusion system. First, we examine the effect of the boundary conditions on the spatially uniform fixed point at locations far from the boundaries by using an asymptotic expansion. We show that, except for a few computational cells adjacent to the boundary, the fixed point practically coincides with the one calculated by using reflective boundary conditions (equivalent to an infinite domain). Next, we introduce a classification of the fixed points based on the wavelength in the infinite‐medium approximation of the system. We use the symbolic manipulator MACSYMA to analytically calculate the amplitude of several such classes of fixed points and we generate bifurcation diagrams for their members. Also, we consider two special classes of periodic solutions; we calculate their amplitude analytically in the infinite‐medium approximation, and generate bifurcation diagrams that shed new light on some previous confusing results. Finally, we present an analysis of fictitious periodic solutions that have been previously reported and incorrectly interpreted.
Numreical solutoin of a class of parabolic partial differential equation arising in optimal control problems with uncertainty
Authors: Dahleh M. , Perice A. P. 
Published: August 2005

AbstractIn this paper the optimal control of uncertain parabolic systems of partial differential equations is investigated. In order to search for controllers that are insensitive to uncertainties in these systems, an iterative optimization procedure is proposed. This procedure involves the solution of a set of operator valued parabolic partial differential equations. The existence and uniqueness of solutions to these operator equations is proved, and a stable numerical algorithm to approximate the uncertain optimal control problem is proposed. The viability of the proposed algorithm is demonstrated by applying it to the control of parabolic systems having two different types of uncertainty.
Analysis of numerical integration in p‐version finite element eigenvalue approximation
Authors: Banerjee Uday , Suri Manil 
Published: August 2005

AbstractWe study the effect of numerical integration when the p‐version of the finite element method is used to approximate the eigenpairs of elliptic partial differential operators. We obtain optimal orders of convergence for approximate eigenvalues and eigenvectors under a certain set of requirements on the quadrature rules employed. This is the same set of conditions that has been shown (in an earlier work) to be sufficient for the optimal approximation of the solutions of the corresponding source problems.
Direct numerical method for an inverse problem of hyperbolic equations
Authors: Lin Tao , Ewing Richard E. 
Published: August 2005

AbstractA coefficient inverse problem of the one‐dimensional hyperbolic equation with overspecified boundary conditions is solved by the finite difference method. The computation is carried out in the x direction instead of the usual t direction. The original boundary condition and the overspecified boundary data are used as the new initial conditions, and the original data at t = 0 are used to compute the coefficient directly. The computation time used by this scheme is almost equal to that for solving the hyperbolic equation in the same region once, even though the inverse problem is essentially nonlinear and hence more difficult to solve. An error estimate is obtained that guarantees the stability of the scheme marching in the x direction. Several numerical experiments are carried out to show the convergence and other properties of the scheme. © 1992 John Wiley & Sons, Inc.
Method of separation of variables and Hamiltonian system
Authors: Wanxie Zhong , Xiangxiang Zhong 
Published: August 2005

AbstractIn the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables ususally leads to the Sturm–Liouville‐type eigenproblems of self‐adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self‐adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite‐dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration. © 1993 John Wiley & Sons, Inc.
Domain‐imbedding alternating direction method for linear elliptic equations on irregular regions using collocation
Authors: Cooper K. D. 
Published: August 2005

AbstractA new method is presented for solving elliptic partial differential equations over two‐dimensional irregular regions. The scheme imbeds the irregular region in a rectangle, and then uses an alternating direction iteration to solve the resulting system of linear equations. Collocation with cubic Hermite splines is used for discretization. The method is shown to be equivalent to a multiboundary alternating direction method. A theory of convergence for a simplified case is given, details of implementation are discussed, and two numerical illustrations are presented. © 1993 John Wiley & Sons, Inc.
Existence and C0, α regularity of minima of a functional related to the grid‐generation problem
Authors: Liao G. , Liu H. 
Published: August 2005

AbstractThe existence of C0,α regularity of minima of a functional arising from the grid‐generation problem are provided in the case of n‐dimensional domains. The functional studied in this article is a linear combination of the length functional, the volume functional, and a higher‐order term. The inclusion of the latter guarantees the existence and regularity of a minimum via the direct method for calculus of variations. © 1993 John Wiley & Sons, Inc.
Linearized stability analysis of staggered‐grid difference schemes for multidimensional viscous incompressible flows
Authors: Kwok Yue‐Kuen , Tam Kin‐Kiu 
Published: August 2005

AbstractThis paper presents a systematic linearized (frozen coefficient) von Neumann stability analysis for various staggered‐grid marker‐and‐cell (MAC) finite difference schemes for solving viscous incompressible fluid flows. These schemes employ the primitive variables formulation and require the velocity field to be divergent free at every time step. It is illustrated that the stability results for staggered‐grid MAC schemes are similar to that for the multidimensional convective‐diffusion equation with constant coefficients. © 1993 John Wiley & Sons, Inc.
Characteristic‐based schemes for dispersive waves I. The method of characteristics for smooth solutions
Authors: Roe Philip L. , Arora Mohit 
Published: August 2005

AbstractIn order to embark on the development of numerical schemes for stiff problems, we have studied a model of relaxing heat flow. To isolate those errors unavoidably associated with discretization, a method of characteristics is developed, containing three free parameters depending on the stiffness ratio. It is shown that such “decoupled” schemes do not take into account the interaction between the wave families and hence result in incorrect wave speeds. We also demonstrate that schemes can differ by up to two orders of magnitude in their rms errors even while maintaining second‐order accuracy. We show that no method of characteristics solution can be better than second‐order accurate. Next, we develop “coupled” schemes which account for the interactions, and here we obtain two additional free parameters. We demonstrate how coupling of the two wave families can be introduced in simple ways and how the results are greatly enhanced by this coupling. Finally, numerical results for several decoupled and coupled schemes are presented, and we observe that dispersion relationships can be a very useful qualitative tool for analysis of numerical algorithms for dispersive waves. © 1993 John Wiley & Sons, Inc.
Dual finite element analysis of three‐dimensional axisymmetric elliptic problems. Part II
Authors: Hlaváček Ivan , Křížek Michal 
Published: August 2005

AbstractIn Part II we derive a priori error estimates for the second type of a pure dual finite element model of a three‐dimensional axisymmetric elliptic problem with mixed boundary conditions. Convergence is proved for both types of model without any regularity assumption upon the solution. An algorithm for finding a particular solution is presented. © 1993 John Wiley & Sons, Inc.
Superconvergence phenomena in nonlinear two‐point boundary‐value problems
Authors: Chow S. S. , Carey G. F. 
Published: August 2005

AbstractSuperconvergence estimates are derived for a class of strongly nonlinear two‐point boundary‐value problems. The analysis considers solution and gradient superconvergence points as well as flux postprocessing formulas. Finally, the extension to parabolic problems is considered. © 1993 John Wiley & Sons, Inc.
Fourth‐order schemes for the wave equation, Maxwell equations, and linearized elastodynamic equations
Authors: Tuomela Jukka 
Published: August 2005

AbstractWith simple finite‐difference operators we construct fourth‐order schemes in space and in time for the wave equation, Maxwell equations, and linearized elastodynamic equations using the modified equation approach. The schemes remain stable for arbitrary heterogeneous mediums because the relevant difference operators are always positive definite. We also present some dispersion curves to show the accuracy of the schemes. © 1994 John Wiley & Sons, Inc.
Two‐dimensional modeling of microscale transport and biotransformation in porous media
Authors: Chen B. , Cunningham A. , Ewing R. , Peralta R. , Visser E. 
Published: August 2005

AbstractWe develop a model for simulating the growth of a biofilm in a tortuous tube. The solutions to the Navier‐Stokes equations and the advection‐diffusion equation are calculated numerically using finite differences. These solutions are then coupled with a biofilm growth model. © 1994 John Wiley & Sons, Inc.
Multidomain direct method and its applicability to the local time‐step concept
Authors: Jelen Jaroslaw A. 
Published: August 2005

AbstractA new multidomain direct (noniterative) method for solving boundary‐value problems is presented. Using this method, the solution is expressed by a linear combination of auxiliary functions and unknowns which pertain to the boundaries of a subdomain. This approach enables us to solve problems independently and exactly, without any iterations between subdomains. As a consequence, different types of equations and coordinate systems may be considered in different subdomains. Moreover, different boundary conditions and variable (in space) time steps may be imposed on the subdomains as well. Applications are given for initial boundary‐value problems with known analytical solutions, including a highly nonlinear porosity equation. © 1994 John Wiley & Sons, Inc.
Improved error estimates for mixed finite‐element approximations for nonlinear parabolic equations: The continuous‐time case
Authors: Garcia Sonia M. F. 
Published: August 2005

AbstractL2‐error estimates are computed for mixed finite‐element methods for second‐order quasilinear (and linear, variable coefficient) parabolic equations. Results are given for the continuous‐time case. The convergence of the values for both the scalar function and the flux is demonstrated. The technique used here covers the lowest‐order Raviart‐Thomas spaces, as well as the higher‐order spaces. A second paper will present the analysis of a fully discrete scheme. © 1994 John Wiley & Sons, Inc.
Finite‐difference methods for one‐dimensional hyperbolic conservation laws
Authors: Berkenbosch A. C. , Kaasschieter E. F. , ten Thije Boonkkamp J. H. M. 
Published: August 2005

AbstractThis article contains a survey of some important finite‐difference methods for one‐dimensional hyperbolic conservation laws. Weak solutions of hyperbolic conservation laws are introduced and the concept of entropy stability is discussed. Furthermore, the Riemann problem for hyperbolic conservation laws is solved. An introduction to finite‐difference methods is given for which important concepts such as, e.g., conservativity, stability, and consistency are introduced. Godunov‐type methods are elaborated for general systems of hyperbolic conservation laws. Finally, flux limiter methods are developed for the scalar nonlinear conservation law. © 1994 John Wiley & Sons, Inc.
A principle of images for absorbing boundary conditions
Authors: Ha Duong T. , Joly P. 
Published: August 2005

AbstractWhen one uses high‐order finite difference schemes for the wave equation, for instance fourth order schemes, the treatment of boundary conditions poses a real difficulty since one needs several additional equations (for the nodes close to the boundary), while one single scalar boundary condition is available. In the case of perfectly reflecting boundary conditions, namely the homogeneous Neumann or Dirichlet conditions, this difficulty can be overcomed by the use of the well‐known image principle, which permits the extension of the equation outside of the domain of calculation by an appropriate symmetrization of the data. We propose in this article a generalization of this principle to the absorbing boundary conditions. Through a symmetrization process, we are led to introduce a damped wave equation with a damping term supported by the boundary. The treatment of the boundary condition is then replaced by the approximation of this new damped wave equation in the whole space. The theoretical justification of our approach is based on new energy estimates for the wave equation (when high‐order absorbing boundary conditions are used), and constitutes an alternative to the use of the well‐known Kreiss criterion to prove the stability of the associated initial boundary value problems. © 1994 John Wiley & Sons, Inc.
Finite‐difference methods for solving the reaction‐diffusion equations of a simple isothermal chemical system
Authors: Twizell E. H. , Wang Yigong , Price W. G. , Fakhr F. 
Published: August 2005

AbstractNumerical methods are proposed for the numerical solution of a system of reaction‐diffusion equations, which model chemical wave propagation. The reaction terms in this system of partial differential equations contain nonlinear expressions. Nevertheless, it is seen that the numerical solution is obtained by solving a linear algebraic system at each time step, as opposed to solving a nonlinear algebraic system, which is often required when integrating nonlinear partial differential equations. The development of each numerical method is made in the light of experience gained in solving the system of ordinary differential equations, which model the well‐stirred analogue of the chemical system. The first‐order numerical methods proposed for the solution of this initialvalue problem are characterized to be implicit. However, in each case it is seen that the numerical solution is obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it is seen that the proposed methods have superior stability properties to those of the well‐known, first‐order, Euler method to which they are compared. Incorporating the proposed methods into the numerical solution of the partial differential equations is seen to lead to two economical and reliable methods, one sequential and one parallel, for solving the travelling‐wave problem. © 1994 John Wiley & Sons, Inc.
Solving multivariable mathematical models by the quadrature and cubature methods
Authors: Civan Faruk 
Published: August 2005

AbstractThe utilization and generalization of quadrature and cubature approximations for numerical solution of mathematical models of multivariable transport processes involving integral, differential, and integro‐differential operators, and for numerical interpolation and extrapolation, are presented. The methodology for determination of the quadrature and cubature weights for composite operators is developed to accommodate for general functional representations. Application of these methods is demonstrated by solving two‐dimensional steady‐state and one‐dimensional transient‐state problems. The solutions are compared with exact‐analytical solutions to evaluate the performance of these methods. It is demonstrated that the quadrature and cubature approximations are simple and universal; i.e., the same formula is applicable irrespective of the order of accuracy of the numerical approximation, the type of linear operator, and the number of temporal and/or spatial variables. Since the quadrature and cubature methods can produce solutions with sufficient accuracy even when using fewer discrete points, both the programming task and computational effort are reduced considerably. Therefore, the quadrature and cubature methods appear to be very practical in solving the mathematical models of a variety of transport processes. © 1994 John Wiley & Sons, Inc.
Spectral Lanczos decomposition method for solving single‐phase fluid flow porous media
Authors: Knizhnerman Leonid , Druskin Vladimir , Liu Qing‐Huo , Kuchuk Fikri J. 
Published: August 2005

AbstractA three‐dimensional well model (r − θ − z) for the simulation of single‐phase fluid flow in porous media is developed. Rather than directly solving the 3‐D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f(A)σ, where A is a real symmetric square matrix and σ is a vector. The linear operator problem is solved by using the spectral Lanczos decomposition method. This formulation gives continuous solutions in time. A 7‐point finite difference scheme is used for the spatial discretization. The model is useful for well testing problems as well as for the simulation of the wireline formation tester tool behavior in heterogeneous reservoirs. The linear operator formulation also permits us to obtain solutions in the Laplace domain, where the wellbore storage and skin can be incorporated analytically. The infinite‐conductivity (uniform pressure) wellbore condition is preserved when mixed boundary conditions, such as partial penetration, occur. The numerical solutions are compared with the analytical solutions for fully and partially penetrated wells in a homogeneous reservoir. © 1994 John Wiley & Sons, Inc.
A mixed finite element method for the Stokes equations
Authors: Farhloul Mohamed 
Published: August 2005

AbstractA new mixed finite element for the Stokes equations is considered. This new finite element is based on a mixed formulation of the Stokes problem in which the gradient of the velocity is introduced and the velocity is approximated by the Raviart‐Thomas element [1]. Optimal error estimates are derived. The number of degrees of freedom, for this element, is the lowest possible, and the local conservation of the mass is assured. A hybrid version of the mixed method is also considered. Finally, some numerical results for the incompressible Navier‐Stokes equations are presented. © 1994 John Wiley & Sons, Inc.
A solution of the convection‐conduction heat‐transfer equation in porous media by the von Rosenberg finite‐difference scheme
Authors: Ginosar Daniel M. , Green Don W. 
Published: August 2005

Abstractvon Rosenberg developed an explicit finite‐difference scheme for solution of the linear convection‐conduction partial differential equation in one space dimension. The method is stable and accurate when a dimensionless ratio of dispersion to convection is between zero and one. In this work, the von Rosenberg method was applied to a linear, one space dimensional set of coupled convection‐conduction equations. The system examined involves the change in temperature resulting from a fluid flowing through a stationary porous solid with heat transfer between the fluid and solid phases. The equations, which describe heat transfer in each phase, were solved simultaneously and, thus, the solution method was required to be implicit rather than explicit. It was observed that when the interphase convective heat‐transfer rate was small relative to the fluid velocity, an implicit solution of the von Rosenberg weighted equations provided a good solution, but when the interphase convective heat‐transfer rate was relatively large, a modified weighting on the equations provided a more accurate, stable solution. © 1994 John Wiley & Sons, Inc.
Nonlinear Poisson‐Boltzmann equation in a model of a scanning tunneling microscope
Authors: Chan Kwong‐Yu , Henderson Douglas , Stenger Frank 
Published: August 2005

AbstractThe nonlinear Poisson‐Boltzmann equation is solved in the region between a sphere and a plane, which models the electrolyte solution interface between the tip and the substrate in a scanning tunneling microscope. A finite difference method is used with the domain transformed into bispherical coordinates. Picard iteration with relaxation is used to achieve convergence for this highly nonlinear problem. An adsorbed molecule on the substrate can also be modelled by a superposition of a perturbing potential in a small region of the plane. An approximate analytical solution using a superposition of individual solutions for plane, the adsorbed molecule, and the sphere is also attempted. Results for cases of different potential values on the boundary surfaces and different distances of the sphere from the plane are presented. The results of the numerical method, the approximate analytical method, as well as the previous solutions of the linearized equation are compared. © 1994 John Wiley & Sons, Inc.
Time‐dependent boundary conditions for the 2‐D linear anisotropic‐viscoelastic wave equation
Authors: Carcione José M. 
Published: August 2005

AbstractWave propagation simulation requires a correct implementation of boundary conditions to avoid numerical instabilities. A boundary treatment based on characteristics, which includes as special cases more simple rheologies involving isotropy and elastic behavior, is applied to the anisotropic‐viscoelastic wave equation. The method introduces the boundary conditions by specifying the values of the incoming variables, which depend on the solution outside the model volume. The formulation ends up with a wave equation for the boundaries that implicitly includes the boundary conditions. The examples illustrate common problems in geophysical modeling, including free surface and nonreflecting conditions. © 1994 John Wiley & Sons, Inc.
An error estimate for a finite volume scheme for a diffusion–convection problem on a triangular mesh
Authors: Herbin Raphaèle 
Published: August 2005

AbstractWe study here a finite volume scheme for a diffusion‐convection equation on an open bounded set Ω of ℝ2, using a triangular mesh for the discretization of Ω. The 4‐point numerical scheme is presented along with the geometrical assumptions on the mesh. An error estimate of order h on the discrete L2 norm is obtained, where h denotes the “size” of the mesh. The proof uses an estimate of order h of the consistency error on the fluxes and an estimate of the number of edges of the mesh between one given triangle and the boundary Ω. © 1995 John Wiley & Sons, Inc.