Showing papers on "Partial differential equation published in 1991"
TL;DR: In this article, the existence results for differential equations with discontinuous right-hand sides with great generality are established and proved for high-order ordinary and partial differential equations for the following problem.
1,881 citations
TL;DR: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied in this paper.
Abstract: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied. It is proved that any monotone, stable, and consistent scheme converges (to the correct solution), provided that there exists a comparison principle for the limiting equation. Several examples are given where the result applies. >
1,063 citations
Book•
12 Jul 1991
TL;DR: The Laplacian Operator in Various Coordinate Systems and some Applications of the Numerical Method of Lines are described, as well as some applications of the ODE and ODE/PDE Applications.
Abstract: What Is the Numerical Method of Lines? Some Applications of the Numerical Method of Lines. Spatial Differentiation. Initial Value Integration. Stability of Numerical Method of Lines Approximations. Additional Applications: Multidimensional Pdes and Adaptive Grids. Appendix A: The Laplacian Operator in Various Coordinate Systems. Appendix B: Spatial Differentiation Routines. Appendix C: Library of ODE and ODE/PDE Applications. Index.
975 citations
TL;DR: In this paper, a comparison of three widely used time propagation algorithms for the time dependent Schrodinger equation is described, and a new method is introduced which is based upon a low-order Lanczos technique.
860 citations
TL;DR: In this article, the authors give three theorems about the existence and uniqueness of mild, strong, and classical solutions of a nonlocal Cauchy problem for a semilinear evolution equation.
831 citations
TL;DR: In this article, uniform estimates and blow-up behavior for solutions of −δ(u) = v(x)eu in two dimensions are presented, with a focus on partial differential equations.
Abstract: (1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.
679 citations
TL;DR: In this article, the natural generalization of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations is discussed. But the natural condition of uralization is not defined.
Abstract: (1991). The natural generalizationj of the natural conditions of ladyzhenskaya and uralľtseva for elliptic equations. Communications in Partial Differential Equations: Vol. 16, No. 2-3, pp. 311-361.
644 citations
TL;DR: A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD), which appears to be more competitive with traditional solution methods.
Abstract: A numerical method, based on a discrete Boltzmann equation, is presented for solving the equations of magnetohydrodynamics (MHD). The algorithm provides advantages similar to the cellular automaton method in that it is local and easily adapted to parallel computing environments. Because of much lower noise levels and less stringent requirements on lattice size, the method appears to be more competitive with traditional solution methods. Examples show that the model accurately reproduces both linear and nonlinear MHD phenomena.
630 citations
Book•
01 Jul 1991
TL;DR: In this paper, a Methode nouvelle a resoudre le probleme de Cauchy for les equations lineaires hyperboliques normales for hyperbolic partial differential equations.
Abstract: Special problems of functional analysis Variational methods in mathematical physics The theory of hyperbolic partial differential equations Comments Appendix: Methode nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales Comments on the appendix Bibliography Index.
451 citations
01 Jan 1991
TL;DR: In this article, the authors present a survey of integrability of PDEs in general and in particular in the context of one-dimensional nonlinear systems, and present an approach to classify them.
Abstract: Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- 1. The Main Ideas in an Illustrative Context.- 2. Survey of Model Equations.- 3. C-Integrable Equations.- 4. Envoi.- Addendum.- References.- Painleve Property and Integrability.- 1. Background.- 1.1 Motivation.- 1.2 History.- 2. Integrability.- 3. Riccati Example.- 4. Balances.- 5. Elliptic Example.- 6. Augmented Manifold.- 7. Argument for Integrability.- 8. Separability.- References.- Integrability.- 1. Integrability.- 2. Introduction to the Method.- 2.1 The WTC Method for Partial Differential Equations.- 2.2 The WTC Method for Ordinary Differential Equations.- 2.3 The Nature of ?.- 2.4 Truncated Versus Non-truncated Expansions.- 3. The Integrable Henon-Heiles System: A New Result.- 3.1 The Lax Pair.- 3.2 The Algebraic Curve and Integration of the Equations of Motion.- 3.3 The Role of the Rational Solutions in the Painleve Expansions.- 4. A Mikhailov and Shabat Example.- 5. Some Comments on the KdV Hierarchy.- 6. Connection with Symmetries and Algebraic Structure.- 7. Integrating the Nonintegrable.- References.- The Symmetry Approach to Classification of Integrable Equations.- 1. Basic Definitions and Notations.- 1.1 Classical and Higher Symmetries.- 1.2 Local Conservation Laws.- 1.3 PDEs and Infinite-Dimensional Dynamical Systems.- 1.4 Transformations.- 2. The Burgers Type Equations.- 2.1 Classification in the Scalar Case.- 2.2 Systems of Burgers Type Equations.- 2.3 Lie Symmetries and Differential Substitutions.- 3. Canonical Conservation Laws.- 3.1 Formal Symmetries.- 3.2 The Case of a Vector Equation.- 3.3 Integrability Conditions.- 4. Integrable Equations.- 4.1 Scalar Third Order Equations.- 4.2 Scalar Fifth Order Equations.- 4.3 Schrodinger Type Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 2.1 The Formal Classical Scattering Matrix in the Solitonless Sector of Rapidly Decreasing Initial Conditions.- 2.2 Infinite-Dimensional Generalization of Poincare's Theorem. Definition of Degenerative Dispersion Laws.- 2.3 Properties of Degenerative Dispersion Laws.- 2.4 Properties of Singular Elements of a Classical Scattering Matrix. Properties of Asymptotic States.- 2.5 The Integrals of Motion.- 2.6 The Integrability Problem in the Periodic Case. Action-Angle Variables.- 3. Applications to Particular Systems.- 3.1 The Derivation of Universal Models.- 3.2 Kadomtsev-Petviashvili and Veselov-Novikov Equations.- 3.3 Davey-Stewartson-Type Equations. The Universality of the Davey-Stewartson Equation in the Scope of Solvable Models.- 3.4 Applications to One-Dimensional Equations.- Appendix I.- Proofs of the Local Theorems (of Uniqueness and Others from Sect.2.3).- Appendix II.- Proof of the Global Theorem for Degenerative Dispersion Laws.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 1.1 Polynomial Mapping of C: What Is Its Integrability?.- 1.2 Commuting Polynomial Mappings of ?N and Simple Lie Algebras.- 1.3 Commuting Rational Mappings of ?Pn.- 1.4 Commuting Cremona Mappings of ?2.- 1.5 Euler-Chasles Correspondences and the Yang-Baxter Equation.- 2. Integrable Lagrangean Mappings with Discrete Time.- 2.1 Hamiltonian Theory.- 2.2 Heisenberg Chain with Classical Spins and the Discrete Analog of the C. Neumann System.- 2.3 The Billiard in Quadrics.- 2.4 The Discrete Analog of the Dynamics of the Top.- 2.5 Connection with the Spectral Theory of the Difference Operators: A Discrete Analogue of the Moser-Trubowitz Isomorphism.- Appendix A.- Appendix B.- References.- The Cauchy Problem for the KdV Equation with Non-Decreasing Initial Data.- 1. Reflectionless Potentials.- 2. Closure of the Sets B(??2).- 3. The Inverse Problem.- References.
396 citations
TL;DR: An algorithm is devised for efficient simulation of waves in excitable media with spatio-temporal resolution and provides accurate solution of the underlying reaction-diffusion equations at small computational cost.
TL;DR: In this article, the authors studied the critical points at infinity of the variational problem, in which the failure of the Palais-Smale condition is the main obstacle for solving equations of type (4).
TL;DR: In this article, the time-dependent, three-dimensional incompressible Navier-Stokes equations are solved in generalized coordinate systems by means of a fractional-step method whose primitive variable formulation uses as dependent variables, in place of the Cartesian components of the velocity: pressure (defined at the center of the computational cell), and volume fluxes across the faces of the cells.
TL;DR: By constraining the potential of the Kadomtsev-Petviashvili (KP) equation to its co-invariants expressed in terms of the squared eigenfunctions, the KP equation is reduced to a (1 + 1)-dimensional system consisting of the generalized multicomponent nonlinear Schrodinger and modified Korteweg-de Vries equations as mentioned in this paper.
TL;DR: In this article, a new approximation scheme for the convective term of the species equations was proposed, which relies on some properties of the exact solution of the Riemann problem for the multi-component system, and applies when an upwind Godunov-type scheme is used for the Euler equations.
TL;DR: In this paper, a numerical method for computing three-dimensional, unsteady incompressible flows is presented, which is a predictor-corrector technique combined with a fractional step method.
TL;DR: The decomposition method can be an effective procedure for solution of nonlinear and/or stochastic continous-time dynamical systems without usual restrictive assumptions as discussed by the authors, which is intended as a convenient tutorial review of the method.
Abstract: The decomposition method can be an effective procedure for solution of nonlinear and/or stochastic continous-time dynamical systems without usual restrictive assumptions. This paper is intended as a convenient tutorial review of the method. 1
TL;DR: In this article, the Riccati equation and the equation for the anharmonic oscillator are expressed in terms of the solutions of a non-integrable nonlinear equation.
TL;DR: A theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms is provided and various numerical approximations of second-order elliptic boundary value problems are applied.
Abstract: We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the \"V cycle, W cycle and a \"V cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way. We apply this theory to various numerical approximations of second-order elliptic boundary value problems. In our first example, we consider certain finite difference multigrid algorithms. In the second example, we consider a finite element multigrid algorithm with nested spaces, which however uses a prolongation operator that does not coincide with the natural subspace imbedding. The third example gives a multigrid algorithm derived from a loosely coupled sequence of approximation grids. Such a loosely coupled grid structure results from the most natural standard finite element application on a domain with curved boundary. The fourth example develops and analyzes a multigrid algorithm for a mixed finite element method using the so-called Raviart-Thomas elements.
TL;DR: The goals of this paper are to carefully define a particular class of well-set incompressible Navier-Stokes problems in the continuum (partial differential equation/PDE) setting and to discuss some relevant and sometimes poorly understood issues related to these well-posed PDE problems, both in the continuity world and in its computer counterpart.
TL;DR: In this article, an adapted pair of process with values in H and K and respectively is defined, which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H.
Abstract: Let K and H be two separable Hilbert spaces and be a cylindrical Wiener process with values in K defined on a probability space denote its natural filtration. Given , we look for an adapted pair of process with values in H and respectively is defined in §1),which solves a semilinear stochastic evolution equation of the backward form: where A is the infinitesimal generators of a C 0-semigroup {eAt } on H. The precise meaning of the equation is A linearized version of that equation appears in infinite-dimensional stochastic optimal control theory as the equation satisfied by the adjoint process. We also give our results to the following backward stochastic partial differential equation:
TL;DR: Galerkin, Galerkin/least-squares and GLSF finite element methods were evaluated by comparing errors pointwise and in integral norms as mentioned in this paper, showing that the GSSF method exhibits superior performance for this class of problems.
Abstract: Finite element methods are presented for the reduced wave equation in unbounded domains. Model problems of radiation with inhomogeneous Neumann boundary conditions, including the effects of a moving acoustic medium, are examined for the entire range of propagation and decay. Exterior boundary conditions for the computational problem over a finite domain are derived from an exact relation between the solution and its derivatives on that boundary. Galerkin, Galerkin/least-squares and Galerkin/gradient least-squares finite element methods are evaluated by comparing errors pointwise and in integral norms. The Galerkin/least-squares method is shown to exhibit superior behavior for this class of problems.
TL;DR: In this paper, the authors studied the symmetry and monotonicity of solutions of fully nonlinear elliptic equations on unbounded domains and showed that they are monotonically and symmetrically symmetric.
Abstract: (1991). Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Communications in Partial Differential Equations: Vol. 16, No. 4-5, pp. 585-615.
TL;DR: Three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations are introduced and the convergence rates of these algorithms are studied.
Abstract: In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported.
TL;DR: In this paper, the problem of finding the most probable model that minimizes the misfit function is formulated in the functional space as opposed to the conventional discrete formulation and solved using iterative gradient methods.
Abstract: Under the assumption that the Earth's thermal field is one-dimensional and purely conductive, the temperature w is related to the Earth model m through a partial differential equation (PDE), where m is the set of model parameters consisting of the ground surface temperature, the background heat flow density, the thermal conductivity, the specific heat capacity, and the rate of heat protection; by setting the time origin sufficiently far in the past, the initial temperature field may be taken as the steady state temperature field. Given data (d0, Cd) on w and a priori information ( m0, Cm) on m, where Cd and Cm are covariance operators describing uncertainties in d0 on m0 respectively, the aim of the least squares inversion is to determine the most probable model that minimizes the misfit function S=1/2〈Cd−1(d-d0),d-d0〉+1/2〈Cm−1(m-m0),m-m0〉. We formulate this problem in the functional space as opposed to the conventional discrete formulation and solve it using iterative gradient methods. The formulation reduces the computation in each iteration to essentially two forward solutions of the PDE, the first for the primal problem: given m, solve for the actual field w, and the second for the dual problem: using the weighted data residuals as heat source, solve for the residual temperature field in the same medium, but with homogeneous boundary conditions and with time reversed. The correlation of the residual and the actual fields, then, gives the gradient and also the Hessian of S, the latter of which evaluated at the most probable model is the approximate a posteriori covariance operator. Because discretization is required only when solving the forward problems, we avoid the computing and storing of partial derivatives of d with respect to discretized m, which can be a prohibitive task when the number of data and the number of discretized m are large.
Book•
01 Jan 1991
TL;DR: In this article, a crash course in distribution theory is presented. But the authors focus on the Dirichlet problem and do not address the problem of initial value problems by Fourier Synthesis.
Abstract: Contents: Power Series Methods.- Some Harmonic Analysis.- Solution of Initial Value Problems by Fourier Synthesis.- Propagators and x-Space Methods.- The Dirichlet Problem.- Appendix: A Crash Course in Distribution Theory.- References.- Index.
TL;DR: In this paper, a similarity reduction of integrable lattices is constructed using the direct linearization method, providing discrete analogues of the Painleve II equation, and an isomonodromic deformation problem for these system is derived.
01 Jan 1991
TL;DR: In this article, the effect of non-parallelism on two-dimensional waves is confirmed to be weak and consequently not responsible for the discrepancies between measurements and theoretical results for parallel flow.
Abstract: We present a new technique for the study of transition in convectively unsta ble flows that employs nonlinear partial diflerential equations of parabolic type based on the slow change of the betsic-flow and the disturbance velocity profiles, wavelength, and growth rate in the streamwise direction. Solutions comparable in accuracy to direct Navier-Stokes simulations can be obtained with a marching procedure utilizing a small fraction of the computationed cost. The development of Tollmien-Schlichting waves in the Blasius boundary layer is investigated. The results are compared with previous work and the effects of nonparaJlelism and nonlinearity are clarified. The effect of nonparallelism on two-dimensional waves is confirmed to be weak and consequently not responsible for the discrepancies between measurements and theoretical results for parallel flow. Possible reasons for the discrepancy are discussed. The effect of basic-flow nonparallelism becomes stronger on oblique waves. While nonlinear effects are small near branch I of the neutral curve, they are significant near branch II and delay or even prevent the decay of the wave. The linearized FSE equations are extended to compressible flow. Results up to a Mach number of 1.6 indicate that compressibility does not alter significantly
TL;DR: In this paper, a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations is presented.
Abstract: The authors present a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations. In order to illustrate this connection they consider the 1D Kuramoto-Sivashinsky equation as a instructive paradigm. They remark that the results presented here apply to many other equations such as the 1D complex Ginzburg-Landau equation, the Chafee-Infante equation, etc.
TL;DR: In this article, the authors generalized the Van Kampen result for the cumulant expansion of a multiplicative stochastic differential equation containing a time-dependent sure matrix to the case of a reactive tracer moving in a heterogeneous aquifer.
Abstract: The cumulant expansion method, used previously by Sposito and Barry (1987) to derive an ensemble average transport equation for a tracer moving in a heterogeneous aquifer, is generalized to the case of a reactive solute that can adsorb linearly and undergo first-order decay. In the process we also generalize the Van Kampen (1987) result for the cumulant expansion of a multiplicative stochastic differential equation containing a time-dependent sure matrix. The resulting partial differential equation exhibits terms with field-scale coefficients that are analogous to those in the corresponding nonstochastic local-scale transport equation. There are also new terms in the third- and fourth-order spatial derivatives of the ensemble average concentration. It is demonstrated that the effective solute velocity for the aqueous concentration, not that for the total concentration (aqueous plus sorbed), is relevant for a field-scale description of solute transport. The field-scale effective solute velocity, dispersion coefficient, retardation factor, and first-order decay parameters, unlike their local-scale counterparts, are time-dependent because of autocorrelations and cross correlations among the random local solute velocity, retardation factor, and first-order decay constant. It is shown also that negative cross correlations between the random tracer solute velocity and the inverse of the local retardation factor may produce both enhanced dispersion and a temporal growth in the field-scale retardation factor. These effects are possible in any heterogeneous aquifer for which a stochastic description of aquifer spatial variability is appropriate.