Abstract: The formulation of a general numerical method for two-dimensional incompressible flow and heat transfer in irregular-shaped domains is presented. The calculation domain is first divided into six-node macroelements. Then each macroelement is divided into four three-node triangular subelements. Polygonal control volumes are associated with the nodes of these elements. All dependent variables other than pressure are stored at the nodes of the subelements, and they are interpolated by functions that respond appropriately to an element Peclet number and the direction of an element-averaged velocity vector. The pressure is stored only at the vertices of the macroelements and is interpolated linearly in these elements. The discretization equations are obtained by deriving algebraic approximations to integral conservation equations applied to the polygonal control volumes. An iterative procedure akin to SIMPLER is used to solve the discretization equations.

Abstract: A numerical method, based on the path-integral formalism, is presented to solve nonlinear Fokker-Planck equations with natural boundary conditions. For one-dimensional stochastic processes, several specific examples possessing exact analytic solutions are evaluated numerically for purposes of comparison. Various discretization prescriptions are investigated and found to be equivalent as expected. The numerical method is shown to give accurate results provided the spatial discretization and the time step satisfy certain relationships determined by the drift and the diffusion functions of the nonlinear Fokker-Planck equations. 26 refs., 5 figs.

Abstract: This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices. These methods have been encoded into our device simulation package which has successfully simulated complex devices in two and three space dimensions. We focus our discussion on nonlinear operator iteration, discretization and scaling procedures, and the efficient solution of the resulting nonlinear and linear algebraic equations. Our companion paper [13] discusses physical aspects of the model equations and presents results from several actual device simulations.

Abstract: In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in En induced by symmetric positive definite matrices Gm. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.

Abstract: Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.

Abstract: In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.

Abstract: A nonlinear parabolic system is derived to describe compressible miscible displace- ment in a porous medium. The system is consistent with the usual model for incompressible miscible displacement. Two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture. The concentration is treated by a Galerkin method in both procedures. while the pressure is treated by either a (lalerkin method or by a parabolic mixed finite element method. Optimal order estimates in L2 and essentially optimal order estimates in Lx are derived for the errors in the approximate solutions for both methods. Introduction. We shall consider the single-phase, miscible displacement of one compressible fluid by another in a porous medium under the assumptions that no volume change results from the mixing of the components and that a pressure-den- sity relation exists for each component in a form that is independent of the mixing. These equations of state will imply that the fluids are in the liquid state. Our model will represent a direct generalization of the model (3), (4), (7) that has been treated extensively for incompressible miscible displacement. The reservoir S will be taken to be of unit thickness and will be identified with a bounded domain in R2. We shall omit gravitational terms for simplicity of exposi- tion; no significant mathematical questions arise when the lower order terms are included.

Abstract: A path-integral solution is derived for processes described by nonlinear Fokker-Planck equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is shown to give accurate results.

Abstract: Consider a linear autonomous system of ordinary differential equations with the property that the norm |U(t)| of each solutionU(t) satisfies |U(t)|?|U(0)| (t?0). We call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations. A givenk-step method is said to be unconditionally contractive if for each stepsizeh>0 the numerical process is contractive. In this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity. It turns out that unconditionally contractive methods are subject to an order barrierp?1. Further the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behaviour of methods with an orderp>1 as well. Most theoretical results in this paper are formulated for differential equations in arbitrary Banach spaces. Applications are given to numerical methods for solving ordinary as well as partial differential equations.

Abstract: Abstract–We discuss a numerical method for determining the flame speed and the structure of freely propagating, adiabatic flames. The method uses a finite difference procedure in which the nonlinear difference equations are solved by a damped, modified, Newton method. This approach is in contrast to the traditional approach of solving a related transient problem until a steady-state solution i5 achieved. Our method is computationally faster than other methods, and it is potentially more accurate because it employs an adaptive gridding strategy. We demonstrate its use for the determination of hydrogen-air flame speeds.

Abstract: The large‐curvature ground‐state model for the transmission coefficient of generalized transition state theory [presented in a previous paper by B C Garrett, D G Truhlar, A F Wagner, and T H Dunning, J Chem Phys 78, 4400(1983)] is tested against accurate quantal calculations of the rate coefficients for collinear reactions with very large inertial effects, namely Cl+HCl→ClH+Cl, Cl+DCl→ClD+Cl, and Cl+MuCl→ClMu+Cl The tests cover the temperature range 200–2400 K The accurate rate calculations are based on reaction probabilities obtained by a new numerical method for solving Schrodinger’s equation in Delves’ coordinates Improved canonical variational transition state theory predicts rate coefficients 50–18 times smaller than those predicted by conventional transition state theory for the H transfer; for the D transfer, the ratio is 20–34; and for Mu it is 22–28×107 The large‐curvature model predicts transmission coefficients as large as 41, 8, and 206 for the H, D, and Mu‐transfer cases at 200 K, decreasing to 12, 11, and 14 at 2400 K Despite the large effect of variationally optimizing the transition state location and the large size of the tunneling effect and the wide variation of both effects with temperature, improved canonical variational transition state theory with large‐curvature ground‐state transmission coefficients (ICVT/LCG) is accurate within a factor of 17 over a temperature range of a factor of 8, 300–2400 K, for all three reactions At 200 K, the ICVT/LCG model underestimates the rate coefficients, by factors of 23, 19, and 15 for H, D, and Mu, respectively

Abstract: Numercal methods are applied in the analysis of coaxial structures used as sensors for in vivo permittivity studies of biological substances. The methods used for the solution of the resulting static conductor-dielectric problems are the Finite Element Method (FEM) and the Method of Moments (MOM) applied to a pair of coupled integral equations. A linear model which relates the sample permittivity to the fringing field capacitance of the sensor is discussed and values of the model parameters are calculated for different types of sensors.

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: A comprehensive analytic model describing current flow in the MIS tunnel junction under steady-state conditions is developed. The tunnel junction is viewed as imposing boundary conditions on the usual set of differential equations governing the electrostatic potential and carrier distributions within the semiconductor. These equations are then solved using the approximation techniques applied in conventional p-n junction theory. Full Fermi-Dirac statistics are used where necessary in the model, and surface states are treated using a Shockley-Read-Hall approach. In computing the band-to-metal tunnel currents, it is assumed that each valley in the conduction band and peak in the valence band can be assigned a single tunneling probability factor describing all transitions between that valley or peak and the metal. On making the above approximations, it is found that the state of the junction is described by two coupled nonlinear algebraic equations, which can be solved by routine iterative techniques. The model is applied to generate current-voltage characteristics for a minority-carrier AI-SiO x - pSi diode, operated both in the dark and as a solar cell, and for a negative barrier AI-SiO x -nSi contact exhibiting photocurrent multiplication. The results obtained are in good agreement with those predicted by more precise numerical methods.

Abstract: We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces

Abstract: Plane and space frames are considered. The member end rotations relative to the chords of the deformed member are assumed to be small. Large translations and rotations of the chord are allowed The longitudinal and transverse displacements are respectively interpolated by linear and cubic functions. The axial strain due to the transverse displacement is averaged over the element length. The averaging is shown to reduce the strain energy and the element stiffness. Without it the element would generally be too stiff. The incremental stiffness matrices are derived in Lagrange coordinates for small rotations. Solution procedures based on a fixed coordinate system and a moving or updated coordinate system are presented. Numerical results indicated that the fixed coordinate procedure works well for small displacement problems. The updated procedure is necessary if large displacements are involved. Comparison of numerical results with those of other methods indicates that the methods presented are competitive.

Abstract: A numerical method involving finite elements is presented for the solution of multidimensional problems of heat transfer with phase transformation. Specific to the method is that both the energy equation in the media and the energy equation on the change of phase interface are regarded as independent governing equations which, when solved through the use of a finite-element formulation, yield the temperature distribution in the media as well as the continuous displacement of the interface. Specific examples illustrate the ability of the method to handle problems with arbitrary boundary conditions which result in irregular two-dimensional shapes of the change of phase interface.

Abstract: Previous studies of the relative convergence (RC) phenomenon in modal analysis showed the need to introduce the edge condition by using an explicit asymptotic behavior of modal coefficients. We do not agree that the edge condition is the only means of uniquely defining a solution. It is demonstrated that the RC can be avoided by including only the bounded feature of modal coefficients. The analytical computations are given in solving the bifurcated parallel plate waveguide. To solve the general problem of convergence of numerical results, a-proof is given of this convergence in modal analysis. The required condition is the use of a "well-conditioned" linear system. With this demonstration, the use of the condition number as purely numerical criterion is justified to ensure the convergence of results.

Abstract: A method for the numerical analysis of rectangular plates based on Mindlin's theory is presented. Any two opposite edges are assumed to be simply supported in the present analysis. A variety of boundary conditions including the mixed and the nonhomogeneous types can be specified along either of the remaining two opposite edges. Numerical results are presented for four examples. The present results clearly show the discrepancies in the results of the usual thin plate theory. Some of the problems associated with the use of the thin plate theory based on Kirchhoff's assumptions are clarified. Finally it is shown that the present segmentation method which is based on the numerical integration of the governing equation system is efficient, economical, reliable, and very accurate in such applications.

Abstract: The weak-guidance approximation, described in Chapter 13, greatly simplifies the determination of the modal fields of optical waveguides, because it depends on solutions of the scalar wave equation, rather than on vector solutions of Maxwell’s equations For circular fibers, with an arbitrary profile, the scalar wave equation must normally be solved by purely numerical methods We discussed the few profiles that have analytical solutions in Chapter 14 These solutions, including those for profiles of practical interest such as the step and clad power-law profiles, are given in terms of special functions or by series expansions, which usually necessitate tables or numerical evaluation to reveal the physical attributes of the modes

Abstract: An accurate and efficient numerical method has been developed for a nonlinear stiff second order two-point boundary value problem. The scheme combines asymptotic methods with the usual solution techniques for two-point boundary value problems. A new modification of Newton's method or quasilinearization is used to reduce the nonlinear problem to a sequence of linear problems. The resultant linear problem is solved by patching local solutions at the knots or equivalently by projecting onto an affine subset constructed from asymptotic expansions. In this way, boundary layers are naturally incorporated into the approximation. An adaptive mesh is employed to achieve an error of $O({1 / {N^2 }}) + O(\sqrt \varepsilon )$. Here, N is the number of intervals and $\varepsilon \ll 1$ is the singular perturbation parameter. Numerical computations are presented.