Showing papers on "Finite difference method published in 1983"
TL;DR: The growth of Gortler vortices in boundary layers on concave walls is investigated in this article, and it is shown that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the gortler problem except for asymptotically small wavelengths.
Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).
402 citations
TL;DR: This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices, and the efficient solution of the resulting nonlinear and linear algebraic equations.
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.
293 citations
TL;DR: In this article, a finite difference procedure that reflects the dominance of convection in incompressible flow in porous media is developed. But this method is not suitable for the case of two-phase, incompressibly flow.
Abstract: Two-phase, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. Convection physically dominates diffusion, and the object of this paper is to develop a finite difference procedure that reflects this dominance. The pressure equation, which is elliptic in appearance, is discretized by a standard five-point difference method. The concentration equation is treated by an implicit finite difference method that applies a form of the method of characteristics to the transport terms. A convergence analysis is given for the method.
177 citations
TL;DR: In this article, new explicit methods for the finite difference solution of a parabolic PDE are derived using stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages.
Abstract: In this paper, new explicit methods for the finite difference solution of a parabolic partial differential equation are derived. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.
154 citations
13 Jul 1983
134 citations
TL;DR: In this paper, a numerical method for determining the flame speed and the structure of freely propagating, adiabatic flames is discussed. But the method is computationally faster than other methods, and it is potentially more accurate because it employs an adaptive gridding strategy.
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.
129 citations
TL;DR: A novel discretization scheme, called "finite boxes," allows an optimal grid-point allocation and can be applied to nonrectangular devices and the advantages and computer resource savings of the new method are described by the simulation of a 100-V diode.
Abstract: A two-dimensional numerical device-simulation system is presented. A novel discretization scheme, called "finite boxes," allows an optimal grid-point allocation and can be applied to nonrectangular devices. The grid is generated automatically according to the specified device geometry. It is adapted automatically during the solution process by equidistributing a weight function which describes the local discretization error. A modified Newton method is used for solving the discretized nonlinear system. To achieve high flexibility the physical parameters can be defined by user-supplied models. This approach requires numerical calculation of parts of the coefficients of the Jacobian. Supplementary algorithms speed up convergence and inhibit the commonly known Newton overshoot. The advantages and computer resource savings of the new method are described by the simulation of a 100-V diode. We also present results for thyristor and GaAs MESFET simulations.
95 citations
TL;DR: In this article, a general class of algorithms for numerical solution of variational inequalities is considered and a convergence proof is given, in particular a multi-grid method for the finite-difference discretization of an obstacle problem for minimal surfaces.
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
70 citations
TL;DR: In this paper, a study of noniterative implicit finite difference methods for hyperbolic systems is presented, and a space-centered method involving two time levels is considered, and various properties such as solvability, stability, dissipation, dispersion, and efficient solution of the algebraic systems are discussed.
Abstract: This paper presents a study of noniterative implicit finite difference methods for hyperbolic systems. New space-centered methods involving two time levels are considered. An analysis is made of various properties such as solvability, stability, dissipation, dispersion, and efficient solution of the algebraic systems. The computation of shock waves with a CFL number larger than the unity is discussed. Unconditional stability results are proved for a case having several space variables. Accurate solutions of the Euler equations are obtained that offer major reductions in computing costs over explicit methods.
61 citations
1 Jan 1983
TL;DR: In this paper, a facility has been constructed to study shear-driven, recirculating flow, namely, a three-dimensional lid-driven cavity flow, and the experimental results are compared to simulations by two different numerical codes, one employing finite differences and one employing infinite elements.
Abstract: A facility has been constructed to study shear-driven, recirculating flow, namely, a three-dimensional lid-driven cavity flow. In the extant case, the cavity depth-to-width aspect ratio is 1:1, while the span-to-width aspect ratio is 3:1. A description of the circulation cell structure obtained by flow visualization is given for two Reynolds numbers with and without temperature-induced density stratification. The experimental results are compared to simulations by two different numerical codes, one employing finite differences and one employing finite elements. The numerical codes reproduce the overall behavior observed in the experiments, but fail to simulate observed longitudinal vortices. There are some significant differences also between the numerical results, highlighting the effects of using the HYBRID (upwind/central) differencing scheme in one code.
51 citations
1 Jan 1983
TL;DR: In this article, a computer program, NASPROP-E, was developed which solves for the flow field surrounding a multibladed propeller and axisymmetric nacelle combination using a finite difference method.
Abstract: To overcome the limitations of classical propeller theory, a computer program, NASPROP-E, was developed which solves for the flow field surrounding a multibladed propeller and axisymmetric nacelle combination using a finite difference method. The governing equations are the three dimensional unsteady Euler equations written in a cylindrical coordinate system. They are marched in time until a steady state solution is obtained. The Euler equations require no special treatment to model the blade work vorticity. The equations are solved using an implicit approximate factorization method. Numerical results are presented which have greatly increased the understanding of high speed propeller flow fields. Numerical results for swirl angle downstream of the propeller and propeller power coefficient are higher than experimental results. The radial variation of coefficient are higher than experimental results. The radial variation of swirl angle, however, is in reasonable agreement with the experimental results. The predicted variation of power coefficient with blade angle agrees very well with data.
TL;DR: In this paper, an explicit finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, non-homogeneous diffusion equation is presented, which possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment.
Abstract: An explicit finite difference approximation procedure which is unconditionally stable for the solution of the general multidimensional, non-homogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e., no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same “marching” type technique of solution. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods. For the examples solved the numerical results obtained by the present method are in satisfactory agreement with the exact solution.
TL;DR: Calculations show that the Galerkin-eigenfunction technique is accurate and in a linear model is clearly computationally more economic than the use of grid boxes through the vertical.
TL;DR: In this article, a modified strongly implicit procedure for solving the system resulting from the modeling of heat conduction in three dimensions is presented, which is derived for a 19 point scheme with the more common 7 point scheme emerging as a special case of the procedure.
Abstract: The application of discretizatio n techniques frequently leads to a system of algebraic equations having a welldefined coefficient structure. A modified strongly implicit procedure for solving the system resulting from the modeling of heat conduction in three dimensions is presented in this work. The method is derived for a 19 point scheme with the more common 7 point scheme emerging as a special case of the procedure. In this way, the asymmetric influence of the additional terms in the LU matrix product is weakened. As a consequence, the method is less sensitive to the iteration parameter and mesh aspect ratio and, in addition, provides considerably more rapid convergence than does the strongly implicit procedure. The increased convergence is exhibited by a significant reduction in the computational cost. The characteristics of the method are examined through application to several model problems and application is made to a more complex three-dimensional problem. Comparisons with the SIP (strongly implicit) and ADI (alternating direction implicit) methods are provided.
TL;DR: In this paper, an algorithm for the solution of advection-diffusion equations based on the finite element method combined with the discretization of the total differential Dρ Dt is presented.
TL;DR: In this article, a lattice truncation scheme for the finite difference time domain approach to the solution of Maxwell's equations has been developed, where the problem space is truncated near the sources and the field components on its boundary are generated from those field values known at retarded times on an interior surface one cell from it with an integral representation of the electromagnetic field.
TL;DR: In this paper, a mathematical model is presented which describes the salt water-fresh water motion with a sharp interface, assuming the validity of the Dupuit approximation, which is used as a base to derive a numerical model (finite difference method) which is unconditionally convergent and stable.
Abstract: A mathematical model is presented which describes the salt water–fresh water motion with a sharp interface, assuming the validity of the Dupuit approximation. This model is used as a base to derive a numerical model (finite difference method) which is unconditionally convergent and stable. A method for solving the equations is selected together with a convergence accelerating procedure. The treatment of the boundary conditions in the interface is discussed, and a general and automatic solution for that problem is presented. Several tests with analytical solutions have been performed with good results.
1 Jan 1983
TL;DR: The use of mixed methods in a two-dimensional finite difference compositional simulator to reduce problems caused by numerical dispersion is presented.
Abstract: Previous studies have shown that standard finite difference techniques cause numerical dispersion and grid orientation problems when used to simulate enhanced recovery processes with adverse mobility ratios. In compositional simulation, numerical dispersion can diffuse sharp fluid interfaces yielding erroneous predictions of fluid compositions and corresponding errors in the velocities of the miscible frontal advance. Numerical dispersion can also effect the computed locations of the boundaries of the regions of single-phase and two-phase flow. Inaccurate fluid velocities and suboptimal use of upstream weighting of transport terms combine to cause many aspects of the numerical dispersion and grid orientation problems. A mixed finite element method has been developed to obtain more accurate approximations to the fluid velocities. In this method the Darcy velocities are considered as primary variables together with the total fluid pressure. Although finite element techniques are used to compute the more accurate fluid velocities, these velocities are then incorporated into a more standard finite difference method for the bulk of the simulation process. This paper presents the use of mixed methods in a two-dimensional finite difference compositional simulator to reduce problems caused by numerical dispersion. Comparisons are made with a standard finite difference simulator on problems involving immiscible displacementmore » and multiple contact miscibility phenomena.« less
TL;DR: Extensions of the adaptive grid finite difference methods to several space dimensions and systems of equations are discussed, achieving substantial improvement in computational efficiency.
15 Nov 1983
TL;DR: A method for calculating nine-point transmissibilities for a general heterogeneous system with unequal grid spacing is presented and previously published methods for calculating these coefficients do not have general applicability to heterogeneous reservoirs and/or various irregular grid spacings.
Abstract: In the past few years there has been increased interest in modeling miscible and thermal recovery processes by the use of reservoir simulation models. The prediction of sharp saturation or temperature fronts resulting from these processes has shown that these simulators can in many cases be very sensitive to grid orientation. One method for minimizing the effect of grid sensitivity is to describe the difference equations with a nine-point rather than the standard five-point finite difference approximations. Inherent in the use of the nine-point approximations is the necessity for calculating consistent transmissibility coefficients. Previously published methods for calculating these coefficients do not have general applicability to heterogeneous reservoirs and/or various irregular grid spacings. This paper presents a method for calculating nine-point transmissibilities for a general heterogeneous system with unequal grid spacing.
TL;DR: Explicit and implicit finite difference methods have been applied to double potential step chronoam-perometry and linear potential sweep voltammetry as discussed by the authors, and the best approximations for each technique have been proposed.
TL;DR: In this paper, the authors considered a two-dimensional horizontal infiltration problem and showed that the maximum error in the L∞ norm could not be made smaller than some positive bound.
TL;DR: The boundary integral equation (BIE) numerical method for linear elastic stress analysis is applied to axisymmetric problems which also involve temperature variations due to steady-state thermal conduction as mentioned in this paper.
Abstract: The boundary integral equation (BIE) numerical method for linear elastic stress analysis is applied to axisymmetric problems which also involve temperature variations due to steady-state thermal conduction. The boundary of the two-dimensional solution domain is discretized into isoparametric quadratic line elements, which provide an excellent modelling capability. Satisfactory agreement is obtained with both exact analytical solutions for test problems and numerical results obtained by the finite difference and finite element methods. The BIE method is applied to the problem of a hollow cylinder with an external semi-circular groove, subjected to axial tensile loading and a temperature difference across the cylinder wall. The results emphasise the very high stress concentrations which can be obtained when the effects of thermal and mechanical loading are superimposed. An important practical advantage of the BIE method is the small amount of labour involved in preparing the mesh data.
TL;DR: In this article, the authors considered the thermal instability of a rotating, heat conducting, micropolar fluid layer heated from below and confined between two rigid boundaries, and the onset of thermal instability was governed by a linear eigenvalue problem.
TL;DR: In this article, three different approaches are employed: the heat balance integral method, the 0-moment integal method, and an implicit finite difference method for heat transfer with ablation in a finite slab.
Abstract: Heat transfer with ablation in a finite slab subjected to time-variant heat fluxes is studied. Three different approaches are employed: the heat balance integral method, the 0-moment integal method, and an implicit finite difference method. Three specific heat fluxes are considered, in terms of linear, exponential, and "power law" functions of time. Numerical results for ablation distance and speed are presented based on the aforementioned techniques. Nomenclature c = heat capacity per unit volume g = a time dependent parameter H = heat of ablation / = index for dimensionless space j = index for dimensionless time k = thermal conductivity L = thickness of the slab TV = total number of elements q0 = dimensional wall heat flux qr = reference heat flux, = k (Tm — T0) IL Q = dimensionless heat flux, = q0L/k( Tm — T0) 5 = ablation distance s - dimensionless ablation distance, = S/L s =dS/dt t = dimensional time
TL;DR: In this paper, a finite difference method was developed for numerical solution of fourth-order parabolic partial differential equations in one and two space variables, which was seen to evolve from a multiderivative method for second-order ordinary differential equations.
TL;DR: In this paper, a fourth-order finite-difference scheme was proposed for discretizing the discrete-ordinates equations for solving numerically the slab transport (Boltzmann) equation.
Abstract: This work is concerned with a theoretical study of a new fourth-order finite-difference scheme for spatially discretizing the discrete-ordinates equations for solving numerically the slab transport (Boltzmann) equation. This analysis considers the quadratic continuous method, whose derivation parallels that of the commonly used diamond difference and linear discontinuous schemes from balance equations for particle conservation across a spatial cell. We provide a convergence analysis of this method and prove that superconvergence phenomena are present for cell-edge and cell-average fluxes. We also present results from an $S_2 $-test problem to show that the asymptotic convergence rates are observed on rather coarse spatial meshes.
TL;DR: In this article, a predictor-corrector scheme was proposed to solve viscous, compressible problems in general coordinates for arbitrary two-dimensional geometries for arbitrary viscous flow regions.
Abstract: In recent years, much progress has been made in solving fluid dynamical problems using finite difference methods. Solving inviscid compressible problems in two and three dimensions has become almost routine with many suitable methods, explicit or implicit, available. The problem of compressible, viscous flows in complicated geometries remains, however, a major challenge. Here fine mesh spacing in the viscous flow regions makes the explicit methods with their simple boundary conditions extremely costly. Existing implicit methods can make use of large time steps, but require inversions of large block tridiagonal matrices. A method recently developed by MacCormack eliminates this disadvantage by introducing a predictor-corrector scheme requiring the inversion of only block bidiagonal matrices. It is the aim of present work to extend this method to allow solution of viscous, compressible problems in general coordinates for arbitrary two-dimensional geometries.