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Showing papers on "Partial differential equation published in 2001"


Journal ArticleDOI
TL;DR: In this paper, a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems is provided, which allows for the understanding and comparison of most of the discontinuous methods that have been proposed over the past three decades.
Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.

3,293 citations


Book
13 Nov 2001
TL;DR: In this article, the authors define trajectories attractors of autonomous and non-autonomous equations as follows: Trajectory attractors are the attractors that follow a trajectory in Hausdorff spaces.
Abstract: Introduction Attractors of autonomous equations: Attractors of autonomous ordinary differential equations Attractors of autonomous partial differential equations Dimension of attractors Attractors of non-autonomous equations: Processes and attractors Translation compact functions Attractors of non-autonomous partial differential equations Semiprocesses and attractors Kernels of processes Kolmogorov $\varepsilon$-entropy of attractors Trajectory attractors: Trajectory attractors of autonomous ordinary differential equations Attractors in Hausdorff spaces Trajectory attractors of autonomous equations Trajectory attractors of autonomous partial differential equations Trajectory attractors of non-autonomous equations Trajectory attractors of non-autonomous partial differential equations Approximation of trajectory attractors Perturbation of trajectory attractors Averaging of attractors of evolution equations with rapidly oscillating terms Proofs of Theorems II.1.4 and II.1.5 Lattices and coverings Bibliography Index.

955 citations


Journal ArticleDOI
TL;DR: New Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations are introduced, based on the use of more precise information about the local speeds of propagation, and are called central-upwind schemes.
Abstract: We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton--Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241--282; A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488; A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720--742]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the one-sided local speeds. This is why we call them central-upwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the Hamilton--Jacobi equations with convex and nonconvex Hamiltonians, and the incompressible Euler and Navier--Stokes equations. The incompressibility condition in the latter equations allows us to treat them both in their conservative and transport form. We apply to these problems the central-upwind schemes, developed separately for each of them, and compute the corresponding numerical solutions.

801 citations


Journal ArticleDOI
TL;DR: In this article, error bounds for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved and the resulting error bounds depend on the number of POD basis functions and on the time discretization.
Abstract: In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.

615 citations


Journal ArticleDOI
TL;DR: In this paper, the effect of adding nonlocal or gradient terms to the constitutive modeling may enhance the ability of the models to describe such situations, and the relation between these enhancements are examined in a continuum damage setting.

590 citations


Book
17 May 2001
TL;DR: In this article, the optimal shape design and partial differential equation for optimal shape optimization for unsteady flows is discussed. But the authors focus on the optimization platform and implementation issues.
Abstract: Introduction 1. Optimal Shape Design 2. Partial Differential Equations for Fluids 3. Some Numerical Methods for Fluids and Examples 4. Automatic Differentiation 5. Optimization Platform and Implementation Issues 6. Consistent Approximations and Approximate Gradients 7. Numerical Results on Shape Optimization 8. Numerical Results on Shape Optimization for Unsteady Flows Index

569 citations


Journal ArticleDOI
TL;DR: In this paper, an efficient method is described to handle mesh indexes in multidimensional problems like numerical integration of partial differential equations, lattice model simulations, and determination of atomic neighbor lists.
Abstract: An efficient method is described to handle mesh indexes in multidimensional problems like numerical integration of partial differential equations, lattice model simulations, and determination of atomic neighbor lists. By creating an extended mesh, beyond the periodic unit cell, the stride in memory between equivalent pairs of mesh points is independent of their position within the cell. This allows to contract the mesh indexes of all dimensions into a single index, avoiding modulo and other implicit index operations.

549 citations


Journal ArticleDOI
TL;DR: In this article, a local radial point interpolation method (LRPIM) is presented to deal with boundary value problems for free vibration analyses of two-dimensional solids, where local weak forms are developed using weighted residual method locally from the partial differential equation of free vibration.

496 citations


Journal ArticleDOI
TL;DR: In this paper, the existence of traveling wave front solutions of reaction-diffusion systems with delay is investigated and a monotone iteration scheme is established for the corresponding wave system.
Abstract: This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model.

487 citations


BookDOI
01 Jan 2001
TL;DR: Holmes et al. as mentioned in this paper proposed a model of low-dimensional models of turbulence, including lattice dynamical systems and extended systems, and three lectures on mathematical fluid mechanics.
Abstract: Preface. Introduction J.C. Robinson, P.A. Glendinning. Spatial correlations and local fluctuations in host-parasite models M.J. Keeling, D.A. Rand. Lattice dynamical systems L.A. Bunimovich (assisted by C. Giberti). Attractors and dynamics in partial differential equations J.K. Hale. Nonlinear dynamics of extended systems P. Collet. Three lectures on mathematical fluid mechanics P. Constantin. Low-dimensional models of turbulence P.J. Holmes, et al.

463 citations


Journal ArticleDOI
TL;DR: A variational formulation is developed which allows, among others, numerical treatment by the finite element method; a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized.

Journal ArticleDOI
TL;DR: The hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form is considered, and an hp-optimal error bound is derived in the hyperbolic case and in the self-adjoint elliptic case.
Abstract: We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by $\frac{1}{2}$ a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

Posted Content
TL;DR: In this article, an algorithm for finding local conservation laws for partial differential equations with any number of independent and dependent variables is presented, which does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations.
Abstract: An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

Reference BookDOI
31 May 2001
TL;DR: Green's Functions for the Wave Equation Heat Equation over Infinite or Semi-Infinite Domains Heat Equations within a Finite Cartesian Domain Heat Exposition within a Cylinder Heat Expositions within a Sphere Product Solution Absolute and Convective Instability Green's functions for the Helmholtz Equation Free-Space Green's Function for the Helms and Poisson's Equation Method of Images Two-Dimensional Poisson Equations over Rectangular and Circular Domains Two-dimensional Problems in a Half-Space Three-dimensional Poisson' Equ
Abstract: Acknowledgments Author Preface List of Definitions Historical Development Mr. Green's Essay Potential Equation Heat Equation Helmholtz's Equation Wave Equation Ordinary Differential Equations Background Material Fourier Transform Laplace Transform Bessel Functions Legendre Polynomials The Dirac Delta Function Green's Formulas What Is a Green's Function? Green's Functions for Ordinary Differential Equations Initial-Value Problems The Superposition Integral Regular Boundary-Value Problems Eigenfunction Expansion for Regular Boundary-Value Problems Singular Boundary-Value Problems Maxwell's Reciprocity Generalized Green's Function Integro-Differential Equations Green's Functions for the Wave Equation One-Dimensional Wave Equation in an Unlimited Domain One-Dimensional Wave Equation on the Interval 0 < x < L Axisymmetric Vibrations of a Circular Membrane Two-Dimensional Wave Equation in an Unlimited Domain Three-Dimensional Wave Equation in an Unlimited Domain Asymmetric Vibrations of a Circular Membrane Thermal Waves Diffraction of a Cylindrical Pulse by a Half-Plane Leaky Modes Water Waves Green's Functions for the Heat Equation Heat Equation over Infinite or Semi-Infinite Domains Heat Equation within a Finite Cartesian Domain Heat Equation within a Cylinder Heat Equation within a Sphere Product Solution Absolute and Convective Instability Green's Functions for the Helmholtz Equation Free-Space Green's Functions for Helmholtz's and Poisson's Equation Method of Images Two-Dimensional Poisson's Equation over Rectangular and Circular Domains Two-Dimensional Helmholtz Equation over Rectangular and Circular Domains Poisson's and Helmholtz's Equations on a Rectangular Strip Three-Dimensional Problems in a Half-Space Three-Dimensional Poisson's Equation in a Cylindrical Domain Poisson's Equation for a Spherical Domain Improving the Convergence Rate of Green's Functions Mixed Boundary Value Problems Numerical Methods Discrete Wavenumber Representation Laplace Transform Method Finite Difference Method Hybrid Method Galerkin Method Evaluation of the Superposition Integral Mixed Boundary Value Problems Appendix: Relationship between Solutions of Helmholtz's and Laplace's Equations in Cylindrical and Spherical Coordinates Answers to Some of the Problems Author Index Subject Index

Journal ArticleDOI
TL;DR: This work analyzes three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions and proves hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal withrespect to p, the degree of polynomial approximation.
Abstract: We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L2 estimates are also derived for the three methods.

Posted Content
TL;DR: In this paper, the authors give a general treatment and proof of the direct conservation law method presented in Part I, which applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form.
Abstract: This paper gives a general treatment and proof of the direct conservation law method presented in Part I. In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

Journal ArticleDOI
TL;DR: In this paper, three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one, leading to the third result.
Abstract: Three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one. The first one states the smoothness of an invariant diffusion density with respect to a parameter. The second establishes a similar smoothness of the solution of the Poisson equation in Rd. The third one states a diffusion approximation result, or in other words an averaging of singularly perturbed diffusion for "fully coupled SDE systems'' or "SDE systems with complete dependence.''

Journal ArticleDOI
TL;DR: In this paper, the surface is represented as the level set of a higher dimensional function and the surface equations are solved in a fixed Cartesian coordinate system using this new embedding function.

Journal ArticleDOI
TL;DR: In this article, the authors present methods for modeling geochemical systems that emphasize the involvement of the gas phase in addition to liquid and solid phases in fluid flow, mass transport, and chemical reactions.
Abstract: Reactive fluid flow and geochemical transport in unsaturated fractured rocks have received increasing attention for studies of contaminant transport, ground- water quality, waste disposal, acid mine drainage remediation, mineral deposits, sedimentary diagenesis, and fluid-rock interactions in hydrothermal systems. This paper presents methods for modeling geochemical systems that emphasize: (1) involvement of the gas phase in addition to liquid and solid phases in fluid flow, mass transport, and chemical reactions; (2) treatment of physically and chemically heterogeneous and fractured rocks, (3) the effect of heat on fluid flow and reaction properties and processes, and (4) the kinetics of fluid-rock interaction. The physical and chemical process model is embodied in a system of partial differential equations for flow and transport, coupled to algebraic equations and ordinary differential equations for chemical interactions. For numerical solution, the continuum equations are discretized in space and time. Space discretization is based on a flexible integral finite difference approach that can use irregular gridding to model geologic structure; time is discretized fully implicitly as a first-order finite difference. Heterogeneous and fractured media are treated with a general multiple interacting continua method that includes double-porosity, dual-permeability, and multi-region models as special cases. A sequential iteration approach is used to treat the coupling between fluid flow and mass transport on the one hand, chemical reactions on the other. Applications of the methods developed here to variably saturated geochemical systems are presented in a companion paper (part 2, this issue).

Journal ArticleDOI
TL;DR: An efficient modification of the Adomian decomposition method is presented that will facilitate the calculations and introduces a promising tool for many linear and nonlinear models.

Book
01 Jan 2001
TL;DR: A local discontinuous Galerkin method for solving KdV type equations containing third derivative terms in one and two space dimensions, which proves L2 stability and a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type in both one and multiple space dimensions.
Abstract: In this paper we develop a local discontinuous Galerkin method for solving KdV type equations containing third derivative terms in one and two space dimensions. The method is based on the framework of the discontinuous Galerkin method for conservation laws and the local discontinuous Galerkin method for viscous equations containing second derivatives; however, the guiding principle for intercell fluxes and nonlinear stability is new. We prove L2 stability and a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type in both one and multiple space dimensions, and we give an error estimate for the linear cases in the one-dimensional case. The stability result holds in the limit case when the coefficients to the third derivative terms vanish; hence the method is especially suitable for problems which are "convection dominated," i.e., those with small second and third derivative terms. Numerical examples are shown to illustrate the capability of this method. The method has the usual advantage of local discontinuous Galerkin methods, namely, it is extremely local and hence efficient for parallel implementations and easy for h-p adaptivity.

Journal ArticleDOI
TL;DR: In this article, a more powerful method to seek exact travelling wave solutions of nonlinear partial differential equations is presented, which uses the good ideas of the extended-tanh function method and our previous method.


Journal ArticleDOI
TL;DR: The error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.
Abstract: We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Journal ArticleDOI
TL;DR: In this paper, a variant of the Eulerian method for two-phase flow that is valid for small particle response time τ is proposed. But it is not suitable for the case of turbophoresis.

Journal ArticleDOI
TL;DR: In this paper, a phase-field-like approach is proposed to simulate liquid-vapor flows with phase change using a 3D continuous medium across which physical properties have strong but continuous variations.

Journal ArticleDOI
TL;DR: It is established that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula, and the flow map fails to be smooth.
Abstract: We establish that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula. As a consequence, the flow map fails to be smooth.

Journal ArticleDOI
TL;DR: In this paper, the general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated.
Abstract: The general conditions under which the quadratic, uniform and monotonic convergence in the quasilinearization method of solving nonlinear ordinary differential equations could be proved are formulated and elaborated. The generalization of the proof to partial differential equations is straight forward. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to different nonlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations. The first few quasilinear iterations already provide extremely accurate and numerically stable answers.

Journal ArticleDOI
TL;DR: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations, and maintains an asymptotically optimal accuracy.
Abstract: A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.

Journal ArticleDOI
TL;DR: Numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort, and a general proof of this fact is impossible.