Abstract: Simple and efficient numerical methods are developed for treating electromagnetic problems of scattering and radiation from surfaces. Special consideration is given to the treatment of edges so that rather arbitrary geometrical configurations may be handled. For the conducting body problems considered, an electric field integral formulation is used, and the method of moments is applied using pulse expansions to represent both the current and the charge. It is demonstrated that proper placement of the current and charge subdomains relative to edges not only is important in treating edges but also yields a convenient numerical procedure. A simple testing scheme is used which is almost as efficient as point-matching. Numerical results indicate that the approach is free of anomalies in the behavior of current near edges and of other previously observed numerical instabilities. Problems considered include conducting strips (both TM and TE), a bent rectangular plate, and both material and conducting bodies of revolution.

Abstract: A general numerical method for convection-diffusion problems is presented The method is formulated for two-dimensional problems, but its key Ideas can be extended to three-dimensional problems The calculation domain is first divided into three-node triangular elements, and then polygonal control volumes are constructed by joining the centroids of the elements to the midpoints of the corresponding sides In each element, the dependent variable is interpolated exponentially in the direction of the element-average velocity vector and linearly in the direction normal to it These interpolation functions respond to an element Peclet number and become linear when it approaches zero The discretization equations are obtained by deriving algebraic approximations to integral conservation equations applied to the polygonal control volumes The proposed method has the conservative property, can handle problems in the whole range of Peclet numbers, and avoids the false-diffusion difficulties that commonly afflict o

Abstract: We shall in this paper consider the problem of computing a generalized solution of a given linear system of equations. The matrix will be partitioned by blocks of rows or blocks of columns. The generalized inverses of the blocks are then used as data to Jacobi- and SOR-types of iterative schemes. It is shown that the methods based on partitioning by rows converge towards the minimum norm solution of a consistent linear system. The column methods converge towards a least squares solution of a given system. For the case with two blocks explicit expressions for the optimal values of the iteration parameters are obtained. Finally an application is given to the linear system that arises from reconstruction of a two-dimensional object by its one-dimensional projections.

Abstract: The relaxation method for solving systems of inequalities is related both to subgradient optimization and to the relaxation methods used in numerical analysis. The convergence theory depends upon two condition numbers. The first one is used mostly for the study of the rate of geometric convergence. The second is used to define a range of values of the relaxation parameter which guarantees finite convergence. In the case of obtuse polyhedra, finite convergence occurs for any value of the relaxation parameter between one and two. Various relationships between the condition numbers and the concept of obtuseness are established.

Abstract: A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(?)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.

Abstract: The stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme

Abstract: A finite element simulation has been carried out for a viscoelastic fluid of the single integral, memory type. In the current work, the memory kernel is chosen to be a single exponential in the time lapse (Maxwell model). However, the formulation is such that it can easily be generalized to more realistic models such as the BKZ theory. From the point of view of numerical analysis, differential models are appealing because they avoid the complexities of memory integrals. However, in these models the viscoelastic effect always enters through terms having the highest-order derivatives. The disadvantage of this situation for numerical analysis appears to be borne out in the experiences reported recently by several workers. In a memory integral formulation, the demand on differentiability of the velocity field is no greater than for the Newtonian fluid. The basic idea in the formulation is the approximation of the memory integral by a Laguerre numerical quadrature formula. The kinematical problem is the computation of the displacement vector from every node to the Laguerre points upstream along particle paths. Since this operation requires the velocity field to be known, the method is restricted to the calculation of non-linear effects as body forces. Thus, the equations being solved in any iteration are those of linear viscous flow with an arbitrary body force. In spite of this limitation, the method converges at dimensionless relaxation times greater than the largest values attained with formulations based on differential models. In the present paper, the method is illustrated with the die entry flow in which the fluid is forced through a four-to-one axisymmetric contraction.

Abstract: A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

Abstract: Two methods have been developed for analyzing MOS transients. One method is analytical and uses the quasi-static approximation. It is useful when the stray capacitance dominates MOS transient performance. The second method is numerical and uses a new boundary value method which can be applied over a wide range of operating speeds. This method includes secondary effects and nonuniform doping, The validity and limits for both methods are verified by comparison with measurements. Transit-time delay and charge-pumping effects are also analyzed using the numerical method. Examples of short-channel behavior of MOS devices are included.

Abstract: An efficient numerical (finite element) method is presented for the dynamic analysis of rapidly propagating cracks in finite bodies, of arbitrary shape, wherein linear-elastic material behavior and two-dimensional conditions prevail. Procedures to embed analytical asymptotic solutions for singularities in stresses/strains near the propagating crack-tip, to account for the spatial movement of these singularities along with the crack-tip, and to directly compute the dynamic stress-intensity factor, are presented. Numerical solutions of several problems and pertinent discussions are presented in Part II of this paper.

Abstract: A nonlinear analysis is carried out for the motion of the inviscid, incompressible fluid in a two-dimensional, rigid, open container which is subjected to forced sinusoidal pitching oscillation. Firstly, the problem is defined as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions. Next, the problem is formulated in the form of a pseudo-variational principle, which provides a basis for our discretization. The finite element method and finite difference method are used spacewise and timewise, respectively. Due to the strong nonlinearity of the problem, an incremental method is used for the numerical analysis. Numerical results obtained by the present method are compared with solutions of the linear theory and experimental data. The difference between linear and nonlinear analysis has been clearly indicated.

Abstract: Approximation concepts and dual method algorithms are combined to create a method for minimum weight design of structural systems. Approximation concepts convert the basic mathematical programming statement of the structural synthesis problem into a sequence of explicit primal problems of separable form. These problems are solved by constructing explicit dual functions, which are maximized subject to nonnegativity constraints on the dual variables. It is shown that the joining together of approximation concepts and dual methods can be viewed as a generalized optimality criteria approach. The dual method is successfully extended to deal with pure discrete and mixed continuous-discrete design variable problems. The power of the method presented is illustrated with numerical results for example problems, including a metallic swept wing and a thin delta wing with fiber composite skins.

Abstract: Part A of this review describes the particular computer-assisted identification service operated by the NCTC. In Part B, the use of probability matrices is examined, discussing various methods of calculating likelihoods and the problems that arise when calculating these from probability matrices. Part C describes the alternative numerical methods of constructing identification keys and the supplementary methods of selecting “best sets” of characters to aid identification. Finally, in Part D, the prospects and limitations of numerical methods in bacterial identification are assessed, first with regard to methodology used and then in terms of performance and practical limitations.

Abstract: Alternating direction implicit (ADI) schemes for two-dimensional parabolic equations with a mixed derivative are constructed by using the class of all $A_0 $-stable linear two-step methods in conjunction with the method of approximate factorization. The mixed derivative is treated with an explicit two-step method which is compatible with an implicit $A_0 $-stable method. The parameter space for which the resulting ADI schemes are second-order accurate and unconditionally stable is determined. Some numerical examples are given.

Abstract: Summary. We present a method to determine the displacement and the stress on the crack plane for a truly threedimensional shear crack of arbitrary shape propagating in an infinite, homogeneous medium which is linearly elastic everywhere off the crack plane. The main idea of the method (which is due to Hamano) is to use a representation theorem in which the displacement at any given point on the crack plane is written as an integral of the traction over the whole crack plane. The tractions are weighted by the threedimensional solution to Lamb’s problem. Such solutions usually require one numerical integration, but fortunately the necessary solutions are obtainable in closed form. The weighting factor is discretized over a space and time grid to solve the integral equation numerically. As a test of the accuracy of our numerical technique, we compare the results with known solutions for two simple cases.

Abstract: : The spectral properties of Jacobi and periodic Jacobi matrices are analyzed and algorithms for the construction of Jacobi and periodic Jacobi matrices with prescribed spectra are presented. Numerical evidence demonstrates that these algorithms are of practical utility. These algorithms have been used in studies of the periodic Toda lattice, and might also be used in studies of inverse eigenvalue problems for Sturm-Louiville equations and Hill's equation. (Author)

Abstract: A three-dimensional numerical model has been developed to study hydrodynamic circulations produced in coastal zones due to tide and wind action. The model consists of a mixed finite-difference/finite element solution of the simplified fluid momentum and continuity equations. A numerical splitting technique is used to reduce the size of model solution matrices while the finite element approach is used over the flow depth to enable irregular sea beds to be tackled easily. Model errors arising from the numerical method are minimized by the use of a Galerkin weighted-residual procedure. The problems associated with modeling the turbulence closure of the basic momentum equations are also investigated with a simplified form of the model and the need for high levels of closure is demonstrated. The potential use of the three-dimensional model is illustrated by prediction of wind-induced flows in Thessaloniki Bay in the Aegean Sea.

Abstract: 1. Introduction.- Summary.- 1.1. Stiffness and Singular Perturbations.- 1.1.1. Motivation.- 1.1.2. Stiffness.- 1.1.3. Singular Perturbations.- 1.1.4. Applications.- 1.2. Review of the Classical Linear Multistep Theory.- 1.2.1. Motivation.- 1.2.2. The Initial Value Problem.- 1.2.3. Linear Multistep Operators.- 1.2.4. Approximate Solutions.- 1.2.5. Examples of Linear Multistep Methods.- 1.2.6. Stability, Consistency and Convergence.- 2. Methods of Absolute Stability.- Summary.- 2.1. Stiff Systems and A-stability.- 2.1.1. Motivation.- 2.1.2. A-stability.- 2.1.3. Examples of A-stable Methods.- 2.1.4. Properties of A-stable Methods.- 2.1.5. A Sufficient Condition for A-stability.- 2.1.6. Applications.- 2.2. Notions of Diminished Absolute Stability.- 2.2.1. A (?)-stability.- 2.2.2. Properties of A(?)-stable Methods.- 2.2.3. Stiff Stability.- 2.3. Solution of the Associated Equations.- 2.3.1. The Problem.- 2.3.2. Conjugate Gradients and Dichotomy.- 2.3.3. Computational Experiments.- 3. Nonlinear Methods.- Summary.- 3.1. Interpolatory Methods.- 3.1.1. Certaine's Method.- 3.1.2. Jain's Method.- 3.2. Runge-Kutta Methods and Rosenbrock Methods.- 3.2.1. Runge-Kutta Methods with v-levels.- 3.2.2. Determination of the Coefficients.- 3.2.3. An Example.- 3.2.4. Semi-explicit Processes and the Method of Rosenbrock.- 3.2.5. A-stability.- 4 Exponential Fitting.- Summary.- 4.1. Exponential Fitting for Linear Multistep Methods.- 4.1.1. Motivation and Examples.- 4.1.2. Minimax fitting.- 4.1.3. An Error Analysis for an Exponentially Fitted F1.- 4.2. Fitting in the Matricial Case.- 4.2.1. The Matricial Multistep Method.- 4.2.2. The Error Equation.- 4.2.3. Solution of the Error Equation.- 4.2.4. Estimate of the Global Error.- 4.2.5. Specification of P.- 4.2.6. Specification of L and R.- 4.2.7. An Example.- 4.3. Exponential Fitting in the Oscillatory Case.- 4.3.1. Failure of the Previous Methods.- 4.3.2. Aliasing.- 4.3.3. An Example of Aliasing.- 4.3.4. Application to Highly Oscillatory Systems.- 4.4. Fitting in the Case of Partial Differential Equations.- 4.4.1. The Problem Treated.- 4.4.2. The Minimization Problem.- 4.4.3. Highly Oscillatory Data.- 4.4.4. Systems.- 4.4.5. Discontinuous Data.- 4.4.6. Computational Experiments.- 5. Methods of Boundary Layer Type.- Summary.- 5.1. The Boundary Layer Numerical Method.- 5.1.1. The Boundary Layer Formalism.- 5.1.2. The Numerical Method.- 5.1.3. An Example.- 5.2. The ?-independent Method.- 5.2.1. Derivation of the Method.- 5.2.2. Computational Experiments.- 5.3. The Extrapolation Method.- 5.3.1. Derivation of the Relaxed Equations.- 5.3.2. Computational Experiments.- 6. The Highly Oscillatory Problem.- Summary.- 6.1. A Two-time Method for the Oscillatory Problem.- 6.1.1. The Model Problem.- 6.1.2. Numerical Solution Concept.- 6.1.3. The Two-time Expansion.- 6.1.4. Formal Expansion Procedure.- 6.1.5. Existence of the Averages and Estimates of the Remainder.- 6.1.6. The Numerical Algorithm.- 6.1.7. Computational Experiments.- 6.2. Algebraic Methods for the Averaging Process.- 6.2.1. Algebraic Characterization of Averaging.- 6.2.2. An Example.- 6.2.3. Preconditioning.- 6.3. Accelerated Computation of Averages and an Extrapolation Method.- 6.3.1. The Multi-time Expansion in the Nonlinear Case.- 6.3.2. Accelerated Computation of $$\bar f$$.- 6.3.3. The Extrapolation Method.- 6.3.4. Computational Experiments: A Linear System.- 6.3.5. Discussion.- 6.4. A Method of Averaging.- 6.4.1. Motivation: Stable Functionals.- 6.4.2. The Problem Treated.- 6.4.3. Choice of Functionals.- 6.4.4. Representers.- 6.4.5. Local Error and Generalized Moment Conditions.- 6.4.6. Stability and Global Error Analysis.- 6.4.7. Examples.- 6.4.8. Computational Experiments.- 4.6.9. The Nonlinear Case and the Case of Systems.- 7. Other Singularly Perturbed Problems.- Summary.- 7.1. Singularly Perturbed Recurrences.- 7.1.1. Introduction and Motivation.- 7.1.2. The Two-time Formalism for Recurrences.- 7.1.3. The Averaging Procedure.- 7.1.4. The Linear Case.- 7.1.5. Additional Applications.- 7.2. Singularly Perturbed Boundary Value Problems.- 7.2.1. Introduction.- 7.2.2. Numerically Exploitable Form of the Connection Theory.- 7.2.3. Description of the Algorithm.- 7.2.4. Computational Experiments.- References.

Abstract: The order of convergence, as the spatial cell widths tend to zero, is derived for six numerical methods that have been proposed for the slab geometry, multigroup, discrete-ordinates neutron transport equations. Our results, which in two cases differ from earlier experimental results, are illustrated by means of a simple test problem.

Abstract: A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.

Abstract: Numerical solutions are presented for the two-dimensional flow past a circular cylinder in an infinite domain. The flow is assumed to be uniform at infinity and the cylinder is allowed to rotate with a constant angular velocity Ω. Ω is chosen to be in the range (0 to 5 W / a ) where a is the radius of the cylinder and W is the mainstream velocity at infinity. To incorporate viscoelastic properties into the flow, an implicit four-constant Oldroyd model is used, and the resulting nonlinear constitutive equations are solved in parallel with the equations of motion as a coupled set of partial differential equations. The method of solution used is a finite difference technique with block over-relaxation. The results are compared with those of other numerical computations as well as with available experimental data. In particular, consideration is given lift experienced by the cylinder and on the streamline patterns and vorticity distribution.

Abstract: An image-restoration method applying the iterative method to solve simultaneous linear equations is described. The advantages of this method are that the memory capacity to be used is minimal, the computation time is very short, and the man-machine interaction in the course of processing is easily effected. Owing to these advantages, this method seems to be superior from a practical viewpoint to other recently proposed linear-algebraic approaches for image restoration. The mathematical basis of this iterative image-restoration method is described and the suitability of this method is presented. The characteristics of this method are clarified through analysis in frequency space. Nonlinear constraints can also be introduced in this method, which restrain occurrence of erratic results caused by noise amplification. Experimental results using a minicomputer-base digital image-processing system demonstrate that the method is very effective and applicable in practice.

Abstract: It is sometimes convenient to express a numerical algorithm in terms of a network model. The physical picture given can often help the engineer to visualize the properties of the method. In field problems, a lumped network model corresponds to a space discrete field while a transmission-line model corresponds to a field which is discrete in space and time. In this paper, the relationship is given between the lumped network models and transmission-line network models used in the steady-state solution of Maxwell's equations in two and three space dimensions. The use of dual networks is also discussed. An analysis is given for the velocity of waves travelling in any direction across the networks and this is used to compare the accuracy of the models. The use of diakoptics or substructures for the solution of large networks is outlined and this is illustrated by a compound two-dimensional example.

Abstract: A rapid and accurate numerical method is presented for the calculation of two-dimension al potential flow in cascades The integral form of continuity is approximated over finite areas in the physical plane, allowing the grid flexibility required by the periodicity, turning, thickness, and stagger of cascades of practical interest, and yielding second-order accuracy on appropriate meshes Excellent agreement between predictions and analytic or measured results are shown for subsonic and transonic test cases A model problem stability analysis supports the experimental observation that the procedure is unconditional ly stable for subsonic flow and conditionally stable for transonic flow