Showing papers on "Discretization published in 1968"

Ray Effects in Discrete Ordinates Equations

Abstract: The nature of anomalous computational effects due to the discretization of the angular variable in transport theory discrete ordinates approximations is described and analyzed. The origin of these ...

An accurate numerical one-dimensional solution of the p-n junction under arbitrary transient conditions

Abstract: A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as mobile carrier and net electric charge densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time. Considerable attention is focused on the numerical analysis of the initial-value-boundary-value problem in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties of the problem, such as stability conditions related to the discretization of partial differential equations of the parabolic type, small differences between nearly equal numbers, and the variation of most quantities over extremely wide ranges within short regions. Results for a particular n + - p single-junction structure under typical external excitations are reported. The iterative scheme of solution for a single device is applicable also to ensembles of active and passive circuit elements. As a simple example, resutls for the combination of an n + - p diode and an external resistor, analyzed under switching conditions, are presented. The inductive behavior of the device for high current pulses, and storage and recovery phenomena under forward-to-reverse bias switching, are also illustrated. ‘Exact’ and conventional approximate analytical results are compared and discrepancies are exposed.

Accurate Numerical Steady-State and Transient One-Dimensional Solutions of Semiconductor Devices

Abstract: Numerical iterative methods of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junctions under both steady-state and transient conditions are presented. The methods are of a very general character: none of the conventional assumptions and restrictions are introduced, and freedom is available in the choice of the doping profile, generation-recombination law, mobility dependencies, injection level., and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage (as a function of time) the solution yields terminal properties and all the quantities of interest in the interior of the device, such as carrier densities, electric field, electrostatic potential, particle and displacement currents, as functions of position (and time). The work is divided into two parts. In Part I a numerical method of solution of the steady-state problem, already available in the literature, is improved and extended, and is applied to a two-contact and a three-contact device. The analytical formulation of the original method is shown to be unsuitable for generating a sound numerical algorithm sufficiently accurate and valid for high reverse bias conditions. Difficulties and limitations are exposed and overcome by an improved formulation extended to any bias condition. As a simple application of the improved formulation, "exact" and first-order theory results for an idealized N-P structure are presented and compared. The poorness of some of the basic assumptions of the conventional first-order theory is exposed, in spite of a satisfactory agreement between the exact and first-order results of the terminal properties for particular bias conditions. Results for an N-P-N transistor are also reported and the inadequacy of the one-dimensional model discussed. The time-dependent analysis of the problem is presented in Part II. The fundamental equations are rearranged to an equivalent set of three non-linear partial differential equations more suitable for numerical methods. A highly non-uniform two-dimensional mesh, subject to maintenance of constant truncation errors in both spatial and time domains of certain pointwise operations, is chosen for the discretization of the problem, in view of the variation of most quantities over extreme ranges within short regions. Consequently an implicit discretization scheme is selected for the second-order partial differential equations of the parabolic type in order to avoid restrictions on the mesh size, without endangering numerical stability. An iterative procedure is necessary at each instant of time to cope with the several non-linearities of the problem and to achieve consistency between the internal distributions and the generating equations. This procedure is easily generalized to incorporate equations pertinent to networks of passive elements and ideal generators connected to the semiconductor device. Results for a particular single-junction structure under typical time-dependent excitations of external current and terminal voltage, and for an N-P diode interacting with an external resistor under switching conditions., are reported and discussed in detail. Considerable attention is focused on the numerical analysis of the steady-state and transient problems in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several difficulties of the problem, such as the small differences between nearly equal numbers, the variation of most quantities over extremely wide ranges in short regions, and the stability conditions related to the discretization of partial differential equations of the parabolic type.

An accurate numerical one-dimensional solution of the p-n junction under arbitrary transient conditions

Abstract: A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as mobile carrier and net electric charge densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time. Considerable attention is focused on the numerical analysis of the initial-value-boundary-value problem in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties of the problem, such as stability conditions related to the discretization of partial differential equations of the parabolic type, small differences between nearly equal numbers, and the variation of most quantities over extremely wide ranges within short regions. Results for a particular n + - p single-junction structure under typical external excitations are reported. The iterative scheme of solution for a single device is applicable also to ensembles of active and passive circuit elements. As a simple example, resutls for the combination of an n + - p diode and an external resistor, analyzed under switching conditions, are presented. The inductive behavior of the device for high current pulses, and storage and recovery phenomena under forward-to-reverse bias switching, are also illustrated. ‘Exact’ and conventional approximate analytical results are compared and discrepancies are exposed.

Error estimate of a fourth-order Runge-Kutta method with only one initial derivative evaluation

Abstract: In the numerical solution of differential equations it is desirable to have estimates of the local discretization (or truncation) errors of solutions at each step. The estimate may be used not only to provide some idea of the errors, but also to indicate when to adjust the step size. If the magnitude of the estimate is greater than the preassigned upper bound, the step size is reduced to achieve smaller local errors. If the magnitude of the estimate is less than the preassigned lower bound, the step size is increased to save the computing time.

Consistency, convergence and stability of general discretizations of the initial value problem

Abstract: Discretizations of nonlinear operators in Banach space are described and the concept of an "inverse" discretization introduced. In the main part of the paper, the very general formalism of BUTCHER for the initial value problem for ordinary differential equations is examined and the sufficiency of conditions for its stability and convergence is demonstrated. The order of convergence of these methods is discussed, and an example is given.

Zur Diskretisierung einer Klasse Fredholmscher Integralgleichungen mit singulärem Wert der Ordnung 1

Abstract: Two analoguous classes of integral equations respectively systems of linear equations, both singular in operator sense, are defined. Linking them by means of a discretisation technique and utilizing the Fredholm theory we prove, that the solution of the associated discrete problem converges rapidly to that of the integral equation. An error bound of the orderO (n q n ),tO < q < 1, is given.