Abstract: Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.

Abstract: The standard model for sea ice dynamics treats the ice pack as a viscous-plastic material that flows plastically under typical stress conditions but behaves as a linear viscous fluid where strain rates are small and the ice becomes nearly rigid. Because of large viscosities in these regions, implicit numerical methods are necessary for timesteps larger than a few seconds. Current solution methods for these equations use iterative relaxation methods, which are time consuming, scale poorly with mesh resolution, and are not well adapted to parallel computation. To remedy this, we have developed and tested two separate methods. First, by demonstrating that the viscous-plastic rheology can be represented by a symmetric, negative definite matrix operator, we have implemented the faster and better behaved preconditioned conjugate gradient method. Second, realizing that only the response of the ice on time scales associated with wind forcing need be accurately resolved, we have modified the model to reduce to the viscous-plastic model at these time scales; at shorter time scales the adjustment process takes place by a numerically efficient elastic wave mechanism. This modification leads to a fully explicit numerical scheme which further improves the computational efficiency and is an advantage for implementations on parallel machines. Furthermore, we observe that the standard viscous-plastic model has poor dynamic response to forcing on a daily time scale, given the standard time step (1 day) used by the ice modeling community. In contrast, the explicit discretization of the elastic wave mechanism allows the elastic-viscous-plastic model to capture the ice response to variations in the imposed stress more accurately. Thus, the elastic-viscous-plastic model provides more accurate results for shorter time scales associated with physical forcing, reproduces viscous-plastic model behavior on longer time scales, and is computationally more efficient. 49 refs., 13 figs., 6 tabs.

Abstract: This book unites the study of dynamical systems and numerical solution of differential equations. The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulted as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

Abstract: Heath 2/e, presents a broad overview of numerical methods for solving all the major problems in scientific computing, including linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, ordinary and partial differential equations, fast Fourier transforms, and random number generators. The treatment is comprehensive yet concise, software-oriented yet compatible with a variety of software packages and programming languages. The book features more than 160 examples, 500 review questions, 240 exercises, and 200 computer problems. Changes for the second edition include: expanded motivational discussions and examples; formal statements of all major algorithms; expanded discussions of existence, uniqueness, and conditioning for each type of problem so that students can recognize "good" and "bad" problem formulations and understand the corresponding quality of results produced; and expanded coverage of several topics, particularly eigenvalues and constrained optimization. The book contains a wealth of material and can be used in a variety of one- or two-term courses in computer science, mathematics, or engineering. Its comprehensiveness and modern perspective, as well as the software pointers provided, also make it a highly useful reference for practicing professionals who need to solve computational problems. Table of contents 1 Scientific Computing 2 Systems of Linear Equations 3 Linear Least Squares 4 Eigenvalues Problems 5 Nonlinear Equations 6 Optimization 7 Interpolation 8 Numerical Integration and Differentiation 9 Initial Value Problems for ODEs 10 Boundary Value Problems for ODEs 11 Partial Differential Equations 12 Fast Fourier Transform 13 Random Numbers and Simulation

Abstract: A number of finite element discretization techniques based on two (or more) subspaces for nonlinear elliptic partial differential equations (PDEs) is presented. Convergence estimates are derived to justify the efficiency of these algorithms. With the new proposed techniques, solving a large class of nonlinear elliptic boundary value problems will not be much more difficult than the solution of one linearized equation. Similar techniques are also used to solve nonsymmetric and/or indefinite linear systems by solving symmetric positive definite (SPD) systems. For the analysis of these two-grid or multigrid methods, optimal ${\cal L}^p$ error estimates are also obtained for the classic finite element discretizations.

Abstract: Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods.

Abstract: A new geometric-optics model has been developed for the calculation of the single-scattering and polarization properties for arbitrarily oriented hexagonal ice crystals. The model uses the ray-tracing technique to solve the near field on the ice crystal surface, which is then transformed to the far field on the basis of the electromagnetic equivalence theorem. From comparisons with the results computed by the finite-difference time domain method, we show that the novel geometric-optics method can be applied to the computation of the extinction cross section and single-scattering albedo for ice crystals with size parameters along the minimum dimension as small as ~6. Overall agreement has also been obtained for the phase function when size parameters along the minimum dimension are larger than ~20. We demonstrate that the present model converges to the conventional ray-tracing method for large size parameters and produces single-scattering results close to those computed by the finite-difference time domain method for size parameters along the minimum dimension smaller than ~20. The present geometric-optics method can therefore bridge the gap between the conventional ray-tracing and the exact numerical methods that are applicable to large and small size parameters, respectively.

Abstract: In this study, a new finite-difference technique is designed to reduce the number of grid points needed in frequency-space domain modeling. The new algorithm uses optimal nine-point operators for the approximation of the Laplacian and the mass acceleration terms. The coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained. This method reduces the number of grid points per wavelength to 4 or less, with consequent reductions of computer memory and CPU time that are factors of tens less than those involved in the conventional second-order approximation formula when a band type solver is used on a scalar machine.

Abstract: A method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix $A$ is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations. It is proved that in exact arithmetic the preconditioner is well defined if $A$ is an H-matrix. The results of numerical experiments are presented.

Abstract: In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.

Abstract: Stochastic differential equations (SDEs) represent physical phenomena dominated by stochastic processes. As for deterministic ordinary differential equations (ODEs), various numerical schemes are proposed for SDEs. In this paper we study the stability of numerical schemes for scalar SDEs with respect to the mean-square norm, which we call $MS$-stability. We will show some figures of the $MS$-stability domain or regions for some numerical schemes and present numerical results which confirm it. This notion is an extension of absolute stability in numerical methods for ODEs.

Abstract: Publisher Summary This chapter explores the numerical methods for solving dynamic programming (DP) problems. The DP framework has been extensively used in economics because it is sufficiently rich to model almost any problem involving sequential decision making over time and under uncertainty. The chapter focuses on continuous Markov decision processes (MDPs) because these problems arise frequently in economic applications. Although, complexity theory suggests a number of useful algorithms, the theory has relatively little to say about important practical issues, such as determining the point at which various exponential-time algorithms such as Chebyshev approximation methods start to blow up, making it optimal to switch to polynomial-time algorithms. In future work, it will be essential to provide numerical comparisons of a broader range of methods over a broader range of test problems, including problems of moderate to high dimensionality.

Abstract: The problem of evaluating linear elastic stress fields in the neighborhood of cracks and notches is considered. An analytical solution valid for cracked and notched components is given in general terms, according to Muskhelishvili's method based on complex stress functions. The solution is particularly useful for V-shape notches in wide and finite plates under uniform tensile loading. It will be demonstrated that some remarkable solutions of fracture mechanics and notch analysis already reported in the literature can be considered special cases of this general solution, as soon as appropriate values of the free parameters are adopted.

Abstract: We present computer simulation data for the effective permittivity (in the quasistatic limit) of a system composed of discrete inhomogeneities of permittivity e1, embedded in a three‐dimensional homogeneous matrix of permittivity e2. The primary purpose of this paper is to study the related issue of the effect of the geometric shape of the components on the dielectric properties of the medium. The secondary purpose is to analyse how the spatial arrangement in these two‐phase materials affects the effective permittivity. The structures considered are periodic lattices of inhomogeneities. The numerical method proceeds by an algorithm based upon the resolution of boundary integral equations. Finally, we compare the prediction of our numerical simulation with the effective medium approach and with results of previous analytical works and numerical experiments.

Abstract: Based on the idea of the DRM, a numerical method has been devised to interpolate the forcing term of partial differential equations by using multiquadric approximations, a special class of radial basis functions, and then use them to approximate particular solutions. To obtain a good shape parameter of the multiquadrics, we use the technique of cross validation. After we find a particular solution, we then use the method of fundamental solutions to solve the homogeneous PDEs. To demonstrate the effectiveness of our method, four numerical results, including a 3D case, are given.

Abstract: Direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that has a high order of truncation and high-resolution characteristics in the evaluation of spatial derivatives. Compact finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial truncation orders. An analytic method with a systematic procedure to achieve maximum resolution characteristics is devised for multidiagonal schemes, based on the idea of the minimization of dispersive (phase) errors in the wave number domain, and these are applied to the analytic optimization of multidiagonal compact schemes. Actual performances of the optimized compact schemes with a variety of truncation orders are compared by means of numerical simulations of simple wave convections, and in this way the most effective compact schemes are found for tridiagonal and pentadiagonal cases, respectively. From these comparisons, the usefulness of an optimized high-order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed. For the optimized high-order spatial schemes, the feasibility of using classical high-order Runge-Kutta time advancing methods is investigated.

Abstract: A new numerical method for the analysis of elastic and elastic-plastic contacts of two rough surfaces has been developed. The method is based on a variational principle in which the real area of contact and contact pressure distribution are those which minimize the total complementary potential energy. The present variational approach guarantees the uniqueness of the solution of the contact problem and significantly reduces the computation time as compared with the conventional matrix inversion method, and thus, makes it feasible to solve 3-D contact problem with large number of contact points. The model is extended to elastic-perfectly plastic contacts. The model is used to predict contact statistics for computer generated surfaces.

Abstract: The nonlinear transformation of wave spectra in shallow water is considered, in particular, the role of wave breaking and the energy transfer among spectral components due to triad interactions Energy dissipation due to wave breaking is formulated in a spectral form, both for energy-density models and complex-amplitude models The spectral breaking function distributes the total rate of random-wave energy dissipation in proportion to the local spectral level, based on experimental results obtained for single-peaked spectra that breaking does not appear to alter the spectral shape significantly The spectral breaking term is incorporated in a set of coupled evolution equations for complex Fourier amplitudes, based on ideal-fluid Boussinesq equations for wave motion The model is used to predict the surface elevations from given complex Fourier amplitudes obtained from measured time records in laboratory experiments at the upwave boundary The model is also used, together with the assumption of random, independent initial phases, to calculate the evolution of the energy spectrum of random waves The results show encouraging agreement with observed surface elevations as well as spectra

Abstract: A new approach to solving the magnetic field integral equation (MFIE) for the current induced on a infinite perfectly conducting rough surface is presented. By splitting the propagator matrix into contributions from the left and from the right of the point of observation, a second kind integral equation can be formed with a new Born term and a new kernel. Following discretization of this new integral equation, the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional LU decomposition; where the time saving factor is roughly N/3 where N is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated. While the new Born term is usually adequate for scattered field calculations, the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems. Results are presented for one-dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40° which illustrate the robustness of the new Born term.