Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of well-posedness, regularity, and solution operator for systems of partial differential equations (PDE’s) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the Rayleigh-Taylor and Kelvin-Helmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) three-wave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wave-number surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described.

Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

Discretization of partial differential equations (PDEs) is based on the theory of function approximation, with several key choices to be made: an integral equation formulation, or approximate solution operator; the type of discretization, defined by the function subspace in which the solution is approximated; the choice of grids, e.g. regular versus irregular grids to conform to the geometry, or static versus solution adaptive grids. We explore some of the common approaches to the choice of form of the PDE and the space-time discretization, leaving discussion of the grids for a later chapter. The goal is to introduce the reader to various forms of discretization and to illustrate the numerical performance of different methods. In particular, we will address how to choose a method that is accurate, robust and efficient for the problem at hand.

Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

The convergence theory for numerical methods approximating time-dependent problems parallels the theory of ordinary differential equations (ODEs) where two types of behavior are studied, namely: (1) the finite time solution and (2) the long-time asymptotic behavior where the solution either passes through an initial transient state and sets into a steady state, or evolves into a periodic or chaotic motion, or escapes to infinity. We describe the notions of consistency, stability, local and global error estimates, resolution and order of accuracy, followed by Lax-Richtmyer equivalence theorem. The rest of the chapter is devoted to practical implications of the convergence theory in terms of the resolution and error estimates together with von Neumann and CFL stability restrictions.

Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

Boundary conditions arise in the process of numerical implementation of given physical boundary conditions on the original physical or truncated domain due to restrictions imposed by computational resources. The latter situation is often encountered when an infinite domain is truncated or remapped onto a finite computational domain. Numerical implementation of the physical or numerical interface boundary conditions should preserve the accuracy and stability of the inner numerical method. The inaccuracies and instabilities created by numerical implementation of the interface and boundary conditions may be localized at the boundaries or interfaces, but more often they may propagate through out the whole computational domain. Examples include: (a) transparent boundary conditions for hyperbolic and dispersive systems, (b) Berenger’s perfectly matched layer (PML) boundary conditions applied to Maxwell’s equations (c) stability analysis in the presence of boundaries and interfaces, and (d) grid interfaces and material interfaces in semi-conductor device simulation models, and finite-difference time-domain (FDTD) discretization of Maxwell’s equations.

Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

Problems with multiple disparate temporal and spatial scales are at the heart of most physical problems, as well as in mathematical modeling problems, in economics and social sciences. Numerical treatment usually allows one to deal with two, or at most, three disparate scales due to limitations of computer resources. Problems with different scales arise when the interest is in the description of the problem on the larger scales, for example the motion of weather fronts in the presence of fast gravity waves; the average trajectory of a fast rotating charged particle in a magnetic field, in which one averages over the fast gyro-period of the particle; the behavior of society incorporating the dynamics of individuals; the role of singularities in solutions of differential equations or the role of geometry; etc. The methods of dealing with such problems are based on either resolving (if ever possible) or not resolving the fine scales. In the latter case, asymptotic analysis of the dominant terms is often used to separate scales or compute the cumulative effect of the small scale motion on the large scale dynamics. For problems where scales are weakly coupled, numerical treatment may often produce physical results as long as the choice of the method has remnants of the smaller scale behavior such as: damping, diffusion, dispersion, etc. For the second type of problem, under-resolved numerical computations usually never produce physical results as cumulative effects are not of the form of the truncation errors, in addition to the fact that the details of the small scale motion are important and have to be accounted for on either a theoretical or experimental basis. In this Chapter we consider examples from stiff ODEs, long-time integrators, Hamiltonian systems, multi-symplectic systems, hyperbolic conservation laws, Godunov methods, Riemann solvers with slope/flux limiters.

Abstract: 2010 Publication year: 2010 Source:Mathematics in Science and Engineering, Volume 213

In this chapter we overview the basic ideas behind numerical grid adaptation to geometrical features of the domain, its interfaces and boundaries, as well as adaptation to the solution features, such as high gradients, that are often employed to improve the accuracy and efficiency of the computation. Through representative examples we discuss implementation and stability issues for static and dynamic remapping grid generation methods. The last two sections describe Level Set and Front Tracking Methods for propagating moving interfaces and fronts.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter discusses the theoretical and practical aspects of computational methods for mathematical modeling of nonlinear systems. A number of computational techniques are considered in the chapter, such as: (1) methods of operator approximation with any given accuracy, (2) operator interpolation techniques including a non-Lagrange interpolation, (3) methods of system representation subject to constraints associated with concepts of causality, memory and stationarity, (4) methods of system representation with an accuracy that is the best within a given class of models, (5) methods of covariance matrix estimation, (6) methods for low-rank matrix approximations, (7) hybrid methods based on a combination of iterative procedures and best operator approximation, and (8) methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter describes the existence theorems for approximating operators of preassigned accuracy and numerical schemes for their practical realization. Nonlinear systems theory is a direct area of applications for the methods of nonlinear operator approximation. The structures of approximating operators considered in the chapter cannot directly extend to the approximation of operators acting on abstract Banach spaces. A number of specific examples are presented for the particular purpose of illustrating the theoretical results. These examples are used as an underlying polynomial algebra, but this is simply a matter of theoretical convenience. While this assumption may not be strictly correct, the errors involved are limited only by machine accuracy, and in principle do not disrupt the analysis. To justify the approximation process, the chapter imposes a basic topological structure and uses the consequent notions of continuity to establish theorems of Weierstrass type.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter discusses the fundamental principles of the general theory for nonlinear operator interpolation. Interpolating operators are naturally connected to the modeling of nonlinear systems. The most widely known formula for interpolation is the formula for real valued functions on the real line, given by Joseph Louis Lagrange. The Lagrange formula is extended to the interpolation of mappings on more general vector spaces. The formula was developed in association with a systematic study of multi-linear mappings that formed the basis of a generalized Weierstrass approximation theorem in Banach space. The weak interpolation can become a strong interpolation in the case of a finite dimensional range space, and when the parameters are chosen appropriately and the output evaluation set is sufficiently fine, it can also provide an approximation to the original mapping in terms of the uniform norm on C([0,1]).

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter discusses the unified approach to finding mathematical models of a realistic dynamical system that represents the system with any pre-assigned accuracy. Constructively defined operators containing special properties give the models. “A realistic dynamical system” means an object with real-world properties, such as causality, memory, stationarity, and so on. Each object can be defined by a legend of historical information. The legend represents the complete history of the object and specifies the state of the object at all times. The current legend specifies the current state. The systematic evolution of state for a collection of real world objects is called a “dynamical system.” The system is specified by specifying each pair of initial and final states. Any such collection of input–output pairs defines a realistic operator. For many centuries, the modeling of real-world objects is a primary interest for both natural science and philosophy.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter discusses the new approaches for the best constructive approximation of nonlinear operators in deterministic and probability spaces. The theory of operator approximation has a direct application to the mathematical modeling of nonlinear systems. Methods of constructive representation of nonlinear systems are a topic of intensive research. The model must approximate the system in a certain sense subject to some restrictions. Such restrictions follow, in particular, from initial known information on the system. For example, in the case of the system transforming deterministic signals, this information can be given by equations describing the signals sets. A number of deep theoretical results related to the investigation of existence, uniqueness, and characterization of elements of the best approximation are established. In many applications, the spaces of images and pre-images are probabilistic and their elements cannot be written analytically. Applications to modeling of nonlinear systems are also discussed in the chapter.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter discusses the different approaches and computational methods for constructing mathematical models for optimal filtering of stochastic signals. The chapter gives wide generalizations of the known Wiener filter to the cases, when an associated linear operator is not invertible, noise is arbitrary, and the filter should satisfy conditions of causality and different types of memory. The chapter outlines a theoretical basis for the description of random vectors with realizations in Banach space. It follows the methods of Halmos, Dunford and Schwartz, and Yosida. Although many of the results are natural extensions of the results for real–valued random variables, the extensions are nontrivial. This material is essential for a proper understanding of the expectation operator. The theory of random vectors in Hilbert space is an extension of the theory of random vectors in Banach space.

Abstract: 2007 Publication year: 2007 Source:Mathematics in Science and Engineering, Volume 212

This chapter describes computational methods for simultaneous data dimensionality reduction and filtering, and subsequent data reconstruction with maximum possible accuracy. Computational methods that are discussed in the chapter are based on the solution of best approximation problems, special iterative procedures and their combination. In signal processing, data dimensionality reduction (often called “compression”) is motivated by a necessity to diminish expenditures for transmission, processing, and storage of large signal arrays. In statistics, data dimensionality reduction is often identified with a procedure for finding so-called principal components of a large random vector—for example, of components of a smaller vector, which preserves principal features of the original vector. The chapter also discusses the standard principal component analysis and Karhunen-Loéve Transform (PCA–KLT). Variations of the degrees of freedom allow for improving the performance of the methods for data processing are also presented in this chapter.