# Showing papers in "Journal of Computational Physics in 1978"

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TL;DR: In this paper, the finite difference methods of Godunov, Hyman, Lax and Wendroff (two-step), MacCormack, Rusanov, the upwind scheme, the hybrid scheme of Harten and Zwas, the antidiffusion method of Boris and Book, and Glimm's method, a random choice method, are discussed.

Abstract: The finite difference methods of Godunov, Hyman, Lax and Wendroff (two-step), MacCormack, Rusanov, the upwind scheme, the hybrid scheme of Harten and Zwas, the antidiffusion method of Boris and Book, the artificial compression method of Harten, and Glimm's method, a random choice method, are discussed. The methods are used to integrate the one-dimensional Eulerian form of the equations of gas dynamics in Cartesian coordinates for an inviscid, nonheat-conducting fluid. The test problem was a typical shock tube problem. The results are compared and demonstrate that Glimm's method has several advantages.

2,448 citations

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TL;DR: A new general purpose algorithm for multidimensional integration is described, an iterative and adaptive Monte Carlo scheme that is considerably more efficient than several others currently in use for a number of sample integrals of high dimension.

Abstract: A new general purpose algorithm for multidimensional integration is described. It is an iterative and adaptive Monte Carlo scheme. The new algorithm is compared with several others currently in use, and shown to be considerably more efficient than all of these for a number of sample integrals of high dimension ( n ⪆ 4 ).

1,348 citations

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TL;DR: A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst and has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX.

Abstract: A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst [1]. This method has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX [2]. These same problems were also solved by various standard iteration methods. On a typical hard problem, the new method is about 8000 times faster than the point Gauss-Seidel method, 200 times faster than the alternating direction implicit method, and 30 times faster than the block successive overrelaxation method with optimum relaxation factor. The new method has two additional virtues. (1) Most of the algorithm is trivially vectorizable with a vector length equal to the full dimension of the system of linear equations. Thus, great savings are possible on vector machines. (2) The new method has a universal scope of application for solution of implicitly differenced partial differential equations. The only restrictions are that the matrix be symmetric and positive definite. The algorithm of Meijerink and van der Vorst which applied only to positive definite symmetric M-matrices is generalized to apply to positive definite symmetric matrices and further generalized to apply to nonsingular matrices arising from partial differential equations. A general description of the method is given. Numerical results are discussed and presented, and an explanation is given for the success of the method.

908 citations

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TL;DR: In this article, the sensitivity analysis presented in this paper is nonlinear and thus permits one to study the effects of large deviations from the nominal parameter values, and since all parameters are varied simultaneously, one can explore regions of parameter space where several parameters deviate simultaneously from their nominal values.

Abstract: Large sets of coupled, nonlinear equations arise in a number of disciplines in connection with computer-based models of physical, social, and economic processes. Solutions for such large systems of equations must be effected by means of digital computers using appropriately designed codes. This paper addresses itself to the critically important problem of how sensitive the solutions are to variations of, or inherent uncertainties in, the parameters of the equation set. We review here, and also present further developments of, our statistical method of sensitivity analysis. The sensitivity analysis presented here is nonlinear and thus permits one to study the effects of large deviations from the nominal parameter values. In addition, since all parameters are varied simultaneously, one can explore regions of parameter space where several parameters deviate simultaneously from their nominal values. We develop here the theory of our method of sensitivity analysis, then detail the method of implementation, and finally present examples of its use.

642 citations

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TL;DR: In this article, a formalism is developed which allows overlap, kinetic energy, potential energy and electron repulsion integrals over cartesian Gaussian functions to be expressed in a very compact form involving easily computed auxiliary functions.

Abstract: A formalism is developed which allows overlap, kinetic energy, potential energy and electron repulsion integrals over cartesian Gaussian functions to be expressed in a very compact form involving easily computed auxiliary functions. Similar formulas involving the same auxiliary functions are given for the common charge moments, electric-field operators, and spin-interaction operators. Recursion relations are given for the auxiliary functions which make possible the use of Gaussian functions of arbitrarily large angular momentum. An algorithm is described for the computation of electron repulsion integrals.

569 citations

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TL;DR: In this paper, the method of collocation using two finite element techniques is applied to the solution of the general population balance equation for particulate systems, and numerical solutions by both techniques are obtained in six cases for which analytical or asymptotic solutions are available.

Abstract: The method of collocation using two finite element techniques is applied to the solution of the general population balance equation for particulate systems. Numerical solutions by both techniques are obtained in six cases for which analytical or asymptotic solutions are available. Errors associated with solving the equation on a finite particle size domain are analyzed. The results indicate that, for simulating particulate system dynamics, both techniques are highly accurate and efficient.

313 citations

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TL;DR: In this paper, a grid free method for approximating incompressible boundary layers is introduced, which is related to the earlier vortex method; simplicity is achieved at the cost of replacing the Navier-Stokes equations by the Prandtl boundary layer equations.

Abstract: A grid free method for approximating incompressible boundary layers is introduced. The computational elements are segments of vortex sheets. The method is related to the earlier vortex method; simplicity is achieved at the cost of replacing the Navier-Stokes equations by the Prandtl boundary layer equations. A new method for generating vorticity at boundaries is also presented; it can be used with the earlier vortex method. The applications presented include (i) flat plate problems, and (ii) a flow problem in a model cylinder-piston assembly, where the new method is used near walls and an improved version of the random choice method is used in the interior. One of the attractive features of the new method is the ease with which it can be incorporated into hybrid algorithms.

295 citations

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Brown University

^{1}TL;DR: In this article, the solutions of the equation u tt − Δu + m 2 u + gu p = 0 for p odd and m, g > 0 were shown to remain bounded as t → ∞.

Abstract: We compute the solutions of the equation u tt − Δu + m 2 u + gu p = 0 for p odd and m , g > 0. Our computations show that (i) the solutions remain bounded as t → ∞, (ii) the amplitude decreases as p increases, and (iii) the number of oscillations increases as p increases. Because of (i), theoretical results imply that the amplitude goes to zero like O(t −3 2) as t → ∞.

262 citations

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TL;DR: In this paper, a method for the numerical calculation of Fourier transforms in variables that are the logarithms of the original variable and transform variable is described, which can also be applied to Bessel and spherical Bessel transforms.

Abstract: A method is described for the numerical calculation of Fourier transforms in variables that are the logarithms of the original variable and transform variable. The method involves only the application of two successive Fourier transforms and can also be applied to Bessel and spherical Bessel transforms. Numerical examples show that the method gives very accurate results up to large values of the transform variable.

234 citations

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TL;DR: In this article, the Voronoi diagram is constructed given a configuration of points, and a procedure for constructing the corresponding Voroni diagram is given, exact for molecules in the bulk polyhedra of surface molecules can be either eliminated or included using a periodic boundary condition.

Abstract: Given a configuration of points, a procedure for constructing the corresponding Voronoi diagram is given The procedure is exact for molecules in the bulk Polyhedra of surface molecules can be either eliminated or included using a periodic boundary condition The construction is of interest in astronomy, biology, chemistry, materials science, as well as in physics (with points representing atoms, molecules, ions, etc) The present method is more efficient than other procedures described in the literature

221 citations

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TL;DR: A computational procedure is outlined for efficient evaluation of four-center coulomb repulsion integrals using contracted Gaussian basis functions and has been incorporated into the GAUSSIAN-70 molecular orbital program.

Abstract: A computational procedure is outlined for efficient evaluation of four-center coulomb repulsion integrals using contracted Gaussian basis functions. By utilizing information common to a shell of basis functions (such as s, px, py, and pz) and by transforming to alternative axes within the contraction loops, this method achieves high efficiency. The technique has been incorporated into the GAUSSIAN-70 molecular orbital program.

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Abstract: A new technique is developed for solving the equations of two-phase fluid dynamics. This technique involves a semi-implicit differencing of the field equations and a variation of the Newton Gauss Seidel iterative method for solving at each time level the resulting system of algebraic equations. Although the technique can be applied to any of several sets of equations representing two-phase flow, including the two-fluid equations, numerical results are presented here for the drift-flux approximation in one dimension. Significant advantages of the method are its stability, ease of programming for complicated flow networks, and ease of extension to problems in two or three dimensions.

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TL;DR: In this paper, Monte Carlo techniques have been used to calculate neutral gas distributions in tokamaks, using track length estimators, suppression of absorption, and splitting with Russian roulette to reduce the variance.

Abstract: Monte Carlo techniques have been used to calculate neutral gas distributions in tokamaks. The algorithm uses track length estimators, suppression of absorption, and splitting with Russian roulette to reduce the variance, so that the algorithm is economic. The resultant package is small in memory requirements and relatively fast ( ∼15 seconds of PDP-10 KI time per neutral density profile). The tokamak is modeled as an infinite cylinder, and the plasma parameters are specified as a function of the radial coordinate of the cylinder. The effects of wall reflection and sputtering yields can be easily computed within the model. The algorithm has been incorporated in a one-dimensional tokamak transport code. The code has also been used to predict the energy spectra of charge-exchange neutrals which form the basis for measurements of ion temperatures in tokamaks.

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TL;DR: In this paper, an exact, computer-oriented Monte Carlo procedure is derived for numerically simulating continuous-time/discrete-state random walks in which the transition probability per unit time from state Sm to state Sn may depend upon the residence time τ in the state Sm. Conditions for applicational feasibility of the simulation procedure are briefly indicated, and explicit stepping algorithms for simple τ-dependencies are obtained.

Abstract: An exact, computer-oriented, Monte Carlo procedure is derived for numerically simulating continuous-time/discrete-state random walks in which the transition probability per unit time from state Sm to state Sn may depend upon the residence time τ in the state Sm. Conditions for applicational feasibility of the simulation procedure are briefly indicated, and explicit stepping algorithms for some simple τ-dependencies are obtained.

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TL;DR: In this paper, the authors trace out the broad characteristics of a class of higher order finite difference schemes which are applicable to the solution of parabolic partial differential equations associated with viscous fluid flow problems.

Abstract: This paper attempts to trace out the broad characteristics of a class of higher order finite difference schemes which are applicable to the solution of parabolic partial differential equations associated with viscous fluid flow problems. The basic method developed here uses the approach of the compact implicit techniques applied to the full spatial operator. The resulting spatial approximation, referred to here as the operator compact implicit method can be implemented with a variety of temporal integration schemes. In particular, a simple factorization technique is employed to resolve higher space dimension problems in terms of simple tridiagonal systems. The operator compact implicit method is compared to standard techniques and to some of the newer compact implicit methods. Stability characteristics, computational efficiency and the results of numerical experiments are discussed.

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TL;DR: In this article, a grid-generation procedure is proposed based on the solution of Laplace's equation for the Cartesian coordinates of the orthogonal grid nodes, and the combined procedure is tested and demonstrated by calculating the flow properties in a diffuser of sufficient divergence to cause recirculation.

Abstract: Many flows of practical interest, such as those that are bounded by curved surfaces, could be calculated in curvilinear coordinates more accurately, conveniently, and economically than in Cartesian coordinates. A calculation procedure is developed by representing the conservation equations in general orthogonal coordinates and so obtaining appropriate finite-difference equations. These equations are written in a similar manner to their Cartesian counterparts, thus enabling the procedure of Gosman and Pun (Imperial College Report HTS/73/2) to be adapted. The viability of such a procedure depends upon the ability to generate an orthogonal grid appropriate to a given flow geometry and consequently a grid-generation procedure is also developed: it is based on the solution, by an iterative finite-difference technique, of Laplace's equation for the Cartesian coordinates of the orthogonal grid nodes. The combined procedures are tested and demonstrated by calculating the flow properties in a diffuser of sufficient divergence to cause recirculation.

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TL;DR: In this article, the authors derive alternate boundary conditions which do prevent wall reflections, and at the same time can be embedded in a natural way in a Galerkin variational formulation of the duct problem.

Abstract: Duct acoustic problems differ sharply from the pure exterior problem in that the classical radiation conditions do not prevent wall reflections in the former as they do in the latter. In this paper we derive alternate boundary conditions which do prevent wall reflections, and at the same time can be embedded in a natural way in a Galerkin variational formulation of the duct problem.

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TL;DR: In this paper, the one-dimensional, time-dependent, multicomponent premixed laminar flame is solved via a highly accurate method of lines approach via the neglect of pressure variations and viscous dissipation and the use of a Lagrangian spatial coordinate.

Abstract: The one-dimensional, time-dependent, multicomponent premixed laminar flame is solved via a highly accurate method of lines approach. The neglect of pressure variations and viscous dissipation and the use of a Lagrangian spatial coordinate reduce the problem to a system of parabolic partial differential equations for the species concentrations and the temperature. Introducing an appropriate B-spline (finite element) basis for the spatial variation and imposing collocation and boundary conditions on the time-dependent coefficients produce a stiff ordinary initial value problem which can be solved by standard techniques. Physical results of special interest include the transient and steady-state profiles of fluid velocity, temperature, and species concentrations through the reaction zone and the upstream velocity (flame speed) of the combustible mixture required to asymptotically stabilize the flame. The analysis is illustrated for the case of an ozone decomposition flame and a comparison with other theoretical predictions shows that the use of less accurate methods can result in significant errors in the predicted values of minor species profiles and the flame speed.

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TL;DR: In this paper, a new class of numerical algorithms for computer simulation of low frequency electromagnetic and electrostatic phenomena in magnetized plasma is presented, which are solved in the limits of quasineutrality and negligible transverse displacement current (Darwin's model).

Abstract: A new class of numerical algorithms for computer simulation of low frequency electromagnetic and electrostatic phenomena in magnetized plasma is presented. Maxwell's equations are solved in the limits of quasineutrality and negligible transverse displacement current (Darwin's model). Electrons are modeled as a fluid with polarization effects ignored. Ions are described as particles. A novel feature of these algorithms is the use of the electron fluid equation of motion to determine the electric field, which renders these numerical schemes remarkably simple and direct. The simulation plasma is either periodic or bounded by particle reflecting conducting walls. Both fully nonlinear codes with spatial grids and linearized gridless codes have been implemented.

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TL;DR: In this paper, a multidimensional hybrid simulation model has been developed for use in studying plasma phenomena on extended time and distance scales, which makes fundamental use of the small Debye length or quasineutrality assumption.

Abstract: A multidimensional hybrid simulation model has been developed for use in studying plasma phenomena on extended time and distance scales. The model makes fundamental use of the small Debye length or quasineutrality assumption. The ions are modeled by particle-in-cell techniques, while the electrons are considered a collision-dominated fluid. Some electron inertial effects are retained. The fields are calculated in the nonradiative Darwin limit. The quasineutral counterpart of Poisson's equation is obtained by first summing the electron and ion momentum equations and then taking the quasineutral limit. The resulting elliptic equation correctly includes those electrostatic potentials which occur in sheath or ambipolar phenomena while neglecting the short-range electrostatic fields which give rise to plasma oscillations. This model has been implemented in a two-dimensional code QN2. A lower hybrid drift unstable equilibrium with parameters accessible to both hybrid and full-particle simulation has been selected as a test of the code and a demonstration of the model. Initial results indicate quite good agreement between the two simulation methods in linear growth rate and wave number.

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TL;DR: A review is presented of the many different finite difference and finite element methods (FDM and FEM) for computing recirculating flows as exemplified by the cavity flow problem and the integrated schemes appear to be simplest and most efficient.

Abstract: A review is presented of the many different finite difference and finite element methods (FDM and FEM) for computing recirculating flows as exemplified by the cavity flow problem. The various methods are categorized according to whether a single integrated system or two segregated, coupled systems are obtained. The integrated schemes appear to be simplest and most efficient, mainly because they satisfy the incompressibility constraint directly in the mean and because the Newton-Raphson method can be used with them. In some cases, the FEM appears to be the most accurate and stable for the same number of unknowns. The method of upwind differencing introduces serious errors in the form of false diffusion which can only be diminished by extreme refinement of mesh sizes. This has to be checked carefully by convergence studies.

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TL;DR: In this paper, the authors systematically evaluate four methods for solving two-dimensional, linear elliptic partial differential equations on general domains: standard finite differences, collocation, Galerkin, and least squares using Hermite cubic piecewise polynomials.

Abstract: We systematically evaluate four methods for solving two-dimensional, linear elliptic partial differential equations on general domains. The four methods are: standard finite differences; collocation, Galerkin, and least squares using Hermite cubic piecewise polynomials. The collocation method is new in that it applies to general curved domains and we describe this aspect in detail. Our test set of 17 problems ranges from simple to moderately complex. The principal conclusion is that collocation is the most efficient method for general use. Standard finite differences is sometimes more efficient for very crude accuracy (where efficiency is not important anyway) but it is also sometimes enormously less efficient even for very modest accuracy. The accuracy of the Galerkin and least-squares methods is sometimes better than collocation, but the extra cost always negates this advantage for our problems.

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TL;DR: This work has devised and tested a practical algorithm for rapidly solving (1) in the difficult case of large n and numerous strong constraints and hopes this will make the distance geometry approach to conformational calculation more feasible for large, highly constrained systems.

Abstract: In conclusion, we have devised and tested a practical algorithm for rapidly solving (1) in the difficult case of large n and numerous strong constraints. Computer time increases only quadratically and memory requirements increase only linearly with n , and there is little difficulty with multiple minima. We hope this will make the distance geometry approach to conformational calculation more feasible for large, highly constrained systems.

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TL;DR: In this paper, a 2 1 2 -dimensional electrostatic particle code in a slab geometry has been developed to study low-frequency oscillations such as drift wave and trapped particle instabilities in a nonuniform bounded plasma.

Abstract: A 2 1 2 - dimensional electrostatic particle code in a slab geometry has been developed to study low-frequency oscillations such as drift wave and trapped particle instabilities in a nonuniform bounded plasma. A drift approximation for the electron transverse motion is made which eliminates the high-frequency oscillations at the electron gyrofrequency and its multiples. It is, therefore, possible to study the nonlinear effects such as the anomalous transport of plasmas within a reasonable computing time using a real mass ratio. Several examples are given to check the validity and usefulness of the model, including those using full electron dynamics.

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Laval University

^{1}TL;DR: In this article, splitting schemes are applied to the numerical solution of a two-dimensional Vlasov equation in configuration space, by treating the convective term and the acceleration term separately.

Abstract: Splitting schemes are applied to the numerical solution of a two-dimensional Vlasov equation. Results obtained when solving the equation in configuration space, by treating the convective term and the acceleration term separately, are compared with results previously obtained using a different method where the two-dimensional Vlasov equation was transformed in velocity space using Hermite polynomials expansion.

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TL;DR: In this paper, the pseudospectral method is applied to the solution of advection/diffusion problems arising from the dispersion of contaminants in the atmosphere, and two techniques are developed for the pseudo-sensor treatment of nonperiodic boundary conditions that typically exist in such problems.

Abstract: The pseudospectral method is applied to the solution of advection/diffusion problems arising from the dispersion of contaminants in the atmosphere. Two techniques are developed for the pseudospectral treatment of nonperiodic boundary conditions that typically exist in such problems. Calculations using the two forms of the pseudospectral method are presented for the dispersion of a contaminant emitted from an elevated, crosswind line source in the atmosphere. The results indicate that pseudospectral methods offer a promising alternative to finite-difference methods for such problems.

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TL;DR: In this paper, a new iteration scheme is proposed for the solution of the ion optics of high current ion beams extracted from a plasma, which requires far less computational effort than other schemes; converges for arbitrary perveance; and is not geometry dependent.

Abstract: A new iteration scheme is proposed for the solution of the ion optics of high current ion beams extracted from a plasma. The scheme (a) requires far less computational effort than other schemes; (b) converges for arbitrary perveance; (c) allows the solution of the problem far back into the extraction plasma and (d) is not geometry dependent, so it should be usable for a wide variety of situations.

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TL;DR: In this paper, a coordinate system suitable for numerical computation of viscous transonic cascade flows is constructed, which consists of coordinate loops surrounding the airfoil and radial coordinate lines normal to the air-foil surface.

Abstract: A coordinate system suitable for the numerical computation of viscous transonic cascade flows is constructed The system consists of coordinate loops surrounding the airfoil and radial coordinate lines normal to the airfoil surface The outermost loop is constructed so that the cascade periodicity conditions can be applied without interpolation between grid points The coordinates are orthogonal on the airfoil surface but gradually become nonorthogonal away from the airfoil The coordinate distribution of mesh points is simple and direct; this is a useful property for the resolution of large solution gradients In addition to the above, the coordinates are generated from discrete input data, little restriction is placed on airfoil camber or spacing, and the entire analysis is easily extended to three dimensions Moreover, the method of coordinate generation can be readily applied to a wide variety of other problems

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TL;DR: In this paper, a convergent numerical model for the calculation of the optical properties of an ion beam is given, which includes the effects of space-charge of the ions and an equilibrium distribution of electrons.

Abstract: A convergent numerical model for the calculation of the optical properties of an ion beam is given. The ion beam is formed by extracting ions from a plasma and subsequently accelerating these ions with an electrode system. The model includes the effects of space-charge of the ions and an equilibrium distribution of electrons. Methods are given for determining the existence of the solution to the nonlinear difference equations, and a convergent iterative numerical procedure is described. Comparisons are made with a procedure that previously has been used to solve such a model.