Abstract: An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.

Abstract: A numerical model is presented allowing calculation of the dc, ac, and large-signal parameters of field-effect transistors (FET's). The numerical procedure is based on finite-difference approximations to the full time-dependent set of equations. The scheme presented uses centered difference quotients and an implicit treatment of the continuity equation. It is shown to be absolutely stable and accurate for time steps below 1 ps. A set of numerical data calculated for one typical example is compared systematically with experimental values. Excellent agreement between measured and computed values is found for the dc characteristics. Small-signal solutions, obtained by Fourier transform methods are also close to the empirical values. The good fit between experiment and numerical simulation is a thorough validation of both the physical model and the numerical procedure.

Abstract: An efficient numerical method for calculating plane, axisymmetric, and fully three-dimensional blunt-body flow is presented. It is a second-order-accurate, time-dependent finite-volume procedure that solves the Euler equations in integral conservation-law form. These equations are written with respect to a Cartesian coordinate system in which an embedded mesh adjusts in time to the motion of the bow shock that is automatically captured as part of the weak solution. With such an adjusting mesh, oscillations in flow properties near the shock are shown to be virtually eliminated. The scheme uses a time-splitting concept that accelerates the convergence appreciably. Comparisons are made between computed and experimental results.

Abstract: This paper presents the use of the finite element method in the solution of thermal transient problems. The thermal transients in a 275 kV oil/paper insulated buried power cable system is calculated when the load current has a prescribed variation with time. The dielectric loss and shield loss are taken into account. The results are presented and commented. The finite element method has definite advantages over the finite difference method for this application.

Abstract: A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.

Abstract: Two classes of high order finite difference methods for first kind Volterra integral equations are constructed. The methods are shown to be convergent and numerically stable.

Abstract: This paper studies the propagation of discretization error for discontinuou ordinary and retarded differential equations. Various applications are given including one which extends a fundamental theorem of Henrici concerning round-off error.

Abstract: EFFICIENT algorithms for the inversion of symmetric tridiagonal matrices are obtained. The results were published in [1]. Tridiagonal matrices are used not only in the application of finite difference methods to boundary value problems for second-order differential equations [2], but also in the solution of problems of nuclear physics [3]. Hence there is great interest in economical methods for the inversion of highorder band matrices by computer. In this paper efficient algorithms for the inversion of symmetric tridiagonal matrices are obtained. The methods of inversion obtained are compared with other methods. The theorem proved is useful for the solution in analytic form of the problem of processing physical information about the motion of charged particles in bubble chambers [4].

Abstract: : In an effort to exploit the favorable stability properties of implicit methods and thereby increase computational efficiency by taking large time steps, an implicit finite-difference method for the multidimensional Navier-Stokes equations is presented. The method is based on a fully-implicit backward time difference scheme which is linearized by Taylor expansion about the known time level to produce a set of coupled linear difference equations which are valid for a given time step. To solve these difference equations, the Douglas-Gunn procedure for generating alternating-direction implicit (ADI) schemes as perturbations of fundamental implicit difference schemes is introduced. The resulting sequence of one-dimensional equations can be solved efficiently by standard block-elimination methods. The method is a one-step method, as opposed to a predictor-corrector method, and requires no iteration to compute the solution for a single time step. The stability and accuracy of the method are examined in a three-dimensional application to subsonic flow in a straight duct with rectangular cross section. (Modified author abstract)

Abstract: The complex nature of heat flow within a modern current-limiting fuselink precludes direct analysis using classical techniques. This paper describes a method which has been developed for determining the prearcing and steady-state behaviour of such devices. It uses a finite difference technique and is numerical, the calculations being performed by a digital computer. Examples of current and temperature distributions found for some fuse-links are given, and comparisons of calculated and test results for fuselink-clearance times are shown.

Abstract: A numerical method is developed to determine the nonlinear dynamic responses of thin, elastic, rectangular plates subjected to pulse-type uniform pressure loads. The nonlinear plate theory used in this study may be identified as the dynamic von Karman theory. The numerical method is based on finite-difference approximations of the differential equations using central difference formulations. A special form of Gaussian elimination is used to solve the system of algebraic equation resulting from the finite-difference formulation. A stability criterion is developed and checked empirically. The convergence of the solution is examined. Four sets of boundary conditions are considered. The use of the method is demonstrated by specific example problems and the results are compared with other approximate solutions.

Abstract: This paper describes an inverse method for designing transonic airfoil sections or for modifying existing profiles. Mixed finite-difference procedures are applied to the equations of transonic small disturbance theory to determine the airfoil shape corresponding to a given surface pressure distribution. The equations are solved for the velocity components in the physical domain and flows with embedded shock waves can be calculated. To facilitate airfoil design, the method allows alternating between inverse and direct calculations to obtain a profile shape that satisfies given geometric constraints. Examples are shown of the application of the technique to improve the performance of several lifting airfoil sections. The extension of the method to three dimensions for designing supercritical wings is also indicated.

Abstract: Publisher Summary This chapter discusses the numerical solution of quasilinear elliptic equations. It describes the numerical solution of nonlinear diffusion equations by implicit finite difference methods. There are certain nonlinear problems that impose severe stability restrictions on explicit methods but not on implicit methods. Notable among these are conductive heat transfer problems with change of phase, the so-called Stefan problems. An implicit method for the diffusion equation means the approximation of the time derivative by a suitable backward difference quotient. It is generally observed that the solution of a parabolic equation by a sequence of elliptic equations is not subject to stability restrictions on the admissible time step. This is an essential feature as in many applications the behavior of a diffusion system has to be modeled over long time periods with locally fine spacial resolution.

Abstract: Two iterative methods for numerically solving the incompressible 2D steady-state Navier-Stokes equation are presented. These are the Numerical Oseen (NOS) method and the Laplacian Driver (LAD) method. Unlike most methods, these are not time-dependent or even time-like in their iterations. The methods make use of recent advances in numerically solving 2D linear second-order partial differential equations with methods which are direct (i.e., non-iterative).

Abstract: An analysis is presented of the error in numerical approximations to a system of elliptic equations describing the steady-state distribution of mobile carriers in a semiconductor device. Although this system has been extensively studied by finite difference methods, the accuracy of the numerical methods employed has not been previously established. Computation schemes are presented for which suitable error estimates are obtained, without assuming an unreasonably small mesh size. In addition, for the one-dimensional problem, the effect of the inexact solution of the discrete equations is estimated.

Abstract: This report contains a description of a numerical technique designed to overcome the difficulties associated with the usual solution of 2-dimensional discretized thermal problems when there are singularities arising from point sources or discontinuous boundary conditions. The method is applied to both the finite-difference and the finite-element approach by incorporating in the numerical computations the known analytical form of the singularities. (16 refs.)

Abstract: A modified multilocal difference scheme is presented for the free vibration analysis of nonhomogeneous orthotropic noncircular cylindrical shells with simply supported curved edges. The problem is formulated in terms of 10 first-order ordinary differential equations and in the finite-difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to more accurate results than those obtained by other multilocal and ordinary difference schemes reported here-to-fore in the literature.

Abstract: The finite-element difference expression was derived by use of the variational principle and finite-element synthesis. Several ordinary finite-difference formulae for the Laplacian term were considered. A particular finite-difference formula for the Laplacian term was chosen to bring the difference expressions of finite-element, finite-difference, and weighted-residuals (Galerkin) methods into the same format. The stability criteria were established for all 3 techniques by use of the general stability, von Neumann, and Dusinberre concepts. The oscillation characteristics were derived for all 3 techniques. The finite-element method is more conservative than the finite-difference method, but not so conservative as the Galerkin method in both stability and oscillation characteristics. (11 refs.)