Downloads & Free Reading Options - Results

This effort studied a proposed numerical method for finding elliptic solutions of the Monge-Ampere equations, in the context of hydrodynamic and antenna design applications. Some computational experiments were performed on representative model problems. A theoretical estimate for the convergence rate of one part of the scheme was obtained. For the problem of reflector antenna design, the solvability of the corresponding linearized problem was demonstrated. A geometric interpretation of the solution was obtained which helps in understanding the problem.

This effort studied a proposed numerical method for finding elliptic solutions of the Monge-Ampere equations, in the context of hydrodynamic and antenna design applications. Some computational experiments were performed on representative model problems. A theoretical estimate for the convergence rate of one part of the scheme was obtained. For the problem of reflector antenna design, the solvability of the corresponding linearized problem was demonstrated. A geometric interpretation of the solution was obtained which helps in understanding the problem.

Numerical Methods for Partial Differential Equations is an active peer-reviewed academic journal covering novel techniques for the numerical solution of partial differential equations (PDEs). It was established in 1985 and is published bi-monthly by John Wiley & Sons, Inc. The journal focuses on the development, analysis, and application of numerical methods for solving PDEs, aiming to advance the field of computational mathematics by providing a platform for researchers to share their expertise, innovative algorithms, and numerical techniques for solving complex PDEs. It covers a wide range of topics, including finite element methods, finite difference methods, spectral methods, meshless methods, and high-performance computing for PDEs. The journal publishes original research articles, computational studies, and software developments related to numerical methods for PDEs, emphasising the theoretical foundation and practical aspects of numerical techniques, highlighting their accuracy, efficiency, stability, and convergence properties. The Internet Archive Collection contains microfilm published between 1985 and 1994 Numerical Methods for Partial Differential Equations can be found online at: onlinelibrary.wiley.com/journal/10982426 The ISSN is: 0749-159X

The research under this effort was concerned with stable high-order methods for nonlinear stiff systems of ordinary differential equations, relaxation methods for large scale circuit analysis, and fast direct methods for elliptic partial differential equations on general regions. More specifically, the convergence of the discretized version of the wave-form relaxation algorithm was shown under suitable assumptions on the stability of the multistep methods employed and on the strength of the feedback. A new large-scale circuit decomposition was shown to be effective for a large class of digital circuits. In the area of fast direct methods for elliptic partial differential equations, a one parameter family of factored discretizations of the Laplace operator was derived. A variant of the marching method was proposed which is much more stable than the conventional approach and is thus applicable to grids with large numbers of discretization steps in each direction.

Uncertainty quantification has been an active research area in the past 15 years because of its potential of significant applications ranging from signal processing to aircraft wing designs. It is well understood that effective numerical methods for stochastic partial differential equations (SPDES) are essential for uncertainty quantification. In the last decade much progress has been made in the construction of numerical algorithms to efficiently solve SPDES with random coefficients and white noise perturbations. However, high dimensionality and low regularity are still the bottleneck in solving real world applicable SPDES with efficient numerical methods. This research is intended to address the numerical analysis as well as algorithm aspects of SPDES. Three major contributions are made in this project: i) Construction and convergence analysis of Quasi Monte Carlo based Particle Swarm Optimization (PSO) method; ii) Efficient adaptive domain sparse grid method for SPDES; iii) High order methods of SPDES via systems of forward backward stochastic differential equations. Our work contains algorithm constructions, rigorous error analysis, and extensive numerical experiments to demonstrate our algorithm efficiency and validity of our theoretical analysis.

The convergence estimates obtained for the Korteweg-de Virus equation have been generalized, under the assumption that the solution u is sufficiently regular. For p 4, it is not known whether a global smooth solution exists corresponding to smooth initial data. It is in fact conjectured that for these cases, the solution may develop a singularity in finite time. A code that uses a spatially and temporally adaptive strategy has been implemented. We are currently investigating the stability of solitary type solutions. As conjectured, these solutions are highly unstable for initial amplitudes larger than one.

The convergence estimates obtained for the Korteweg-de Virus equation have been generalized, under the assumption that the solution u is sufficiently regular. For p 4, it is not known whether a global smooth solution exists corresponding to smooth initial data. It is in fact conjectured that for these cases, the solution may develop a singularity in finite time. A code that uses a spatially and temporally adaptive strategy has been implemented. We are currently investigating the stability of solitary type solutions. As conjectured, these solutions are highly unstable for initial amplitudes larger than one.

The convergence estimates obtained for the Korteweg-de Virus equation have been generalized, under the assumption that the solution u is sufficiently regular. For p 4, it is not known whether a global smooth solution exists corresponding to smooth initial data. It is in fact conjectured that for these cases, the solution may develop a singularity in finite time. A code that uses a spatially and temporally adaptive strategy has been implemented. We are currently investigating the stability of solitary type solutions. As conjectured, these solutions are highly unstable for initial amplitudes larger than one.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigations by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigations by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigations by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.

The physical phenomena described by nonlinear partial differential equations have become at present the central theme of investigation by many researchers. A good understanding of most physical processes requires accounting for nonlinear effects and, consequently, methods for studying nonlinear equations have to be developed. Among nonlinear equations the Dirichlet problem for the Monge-Ampere equation is the model case for fully nonlinear equations.

The authors have studied a novel numerical solution approach to Monge Ampere type equations. The equations have important applications. In particular, the leading term in the balance equation in dynamic meteorology has the form (1) . In a more complicated form, an equation of this type appears in the von Karman system of equations for elasticity and also in inverse problems of geometric optics. It also turned out that recent progress in the study of fully nonlinear equations became possible after important properties of the equation (1) were discovered.

The authors have studied a novel numerical solution approach to Monge Ampere type equations. The equations have important applications. In particular, the leading term in the balance equation in dynamic meteorology has the form (1) . In a more complicated form, an equation of this type appears in the von Karman system of equations for elasticity and also in inverse problems of geometric optics. It also turned out that recent progress in the study of fully nonlinear equations became possible after important properties of the equation (1) were discovered.

We derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds. These techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude. Our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales, some of which are sometimes treated exactly. The efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator. This is analyzed and a technique is discussed for reducing this error through a simple change of variables. A procedure for generating the appropriate boundary data for the intermediate solutions is also presented. (Author)

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

The focus of research was the application of systolic array architectures to computations in numerical linear algebra, and the applications of these architectures to digital signal processing and elliptic partial differential equations (PDE). Research was conducted on a number of topics: Matrix triangularization by systolic arrays; Singular value and eigenvalue computations; Elliptic PDE; and Updating Cholesky factorizations.

A survey of numerical methods for time dependent partial differential equations is presented. The emphasis is on practical applications to large scale problems. A discussion of new developments in high order methods and moving grids is given. The importance of boundary conditions is stressed for both internal and external flows. A description of implicit methods is presented including generalizations to multidimensions. Shocks, aerodynamics, meteorology, plasma physics and combustion applications are also briefly described.