Showing papers on "Numerical analysis published in 1983"
TL;DR: In this paper, the authors used mesh refnement and extrapolation to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls.
Abstract: Details are given of the computational method used to obtain an accurate solution of the equations describing two-dimensional natural convection in a square cavity with differentially heated side walls. Second-order, central difference approximations were used. Mesh refnement and extrapolation led to solutions for 103⩽Ra⩽10 6 which are believed to be accurate to better than 1 per cent at the highest Rayleigh number and down to one-tenth of that at the lowest value.
2,529 citations
TL;DR: In this paper, an efficient and reliable numerical technique of high-order accuracy is presented for solving problems of steady viscous incompressible flow in the plane, and is used to obtain accurate solutions for the driven cavity.
492 citations
01 Jan 1983
TL;DR: A modification of the mixed method is proposed when the flow is located at sources and sinks (i.e., wells), and convergence is established at reduced rates in the special case when the viscosity of the mixture is independent of the concentration.
Abstract: The miscible displacement ofone incompressible fluid by another in a porous medium is governedby a System oftwo equations, one ofelliptic formfor the pressure and the other of par abolie formfor the concentration ofone ofthefluids. The pressure appears in the concentration only through its velocity field, and it is appropriate to choose a numerical method that approximates the velocity directly. The pressure is approximated by a mixed finite element method and the concentration by a standard Galerkin method. Optimal order estimâtes are derived when the imposed externalflows are smoothly distributed. A modification of the mixed method is proposed when the flow is located at sources and sinks (i.e., wells), and convergence is established at reduced rates in the special case when the viscosity of the mixture is independent of the concentration.
343 citations
TL;DR: In this paper, a general numerical method for two-dimensional incompressible flow and heat transfer in irregular-shaped domains is presented, where the calculation domain is first divided into six-node macroelements, and each macroelement is divided into four three-node triangular subelements.
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.
313 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.
278 citations
TL;DR: A general iterative scheme for the numerical solution of finite dimensional variational inequalities that contains the projection, linear approximation and relaxation methods but also induces new algorithms and allows the possibility of adjusting the norm at each step of the algorithm.
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.
271 citations
TL;DR: In this paper, a numerical method based on the path-integral formalism is presented to solve nonlinear Fokker-Planck equations with natural boundary conditions, which 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.
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.
234 citations
TL;DR: An iterative algorithm for the solution of the Helmholtz equation is developed based on a preconditioned conjugate gradient iteration for the normal equations.
231 citations
TL;DR: In this article, a numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems, including inviscid and viscous Burgers equations.
222 citations
TL;DR: In this paper, the authors investigated the behavior of Kaczmarz's method with relaxation for inconsistent systems and showed that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system.
Abstract: We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.
217 citations
TL;DR: In this article, a new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented, which uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step.
TL;DR: In this paper, an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations is presented.
Abstract: The Osher algorithm for solving the Euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations. A first-order accurate version of the numerical method was derived by Osher circa 1980 for the one-dimensional non-isentropic Euler equations in Cartesian coordinates. In this paper, the extension of the scheme to arbitrary two-dimensional geometries is explained. Results are then presented for several example problems in one and two dimensions. Future work will include extension of the method to second-order accuracy and the development of implicit time differencing for the Osher algorithm.
TL;DR: In this article, the authors analyzed the behavior of the so-called p-version of the finite element method when applied to the equations of plane strain linear elasticity and established optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[.
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.
TL;DR: In this article, a nonlinear parabolic system is derived to describe compressible miscible displace- ment in a porous medium and two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture.
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.
TL;DR: In this paper, a general theory is given which yields necessary and sufficient conditions for unconditional contractivity, and the concept of a contractivity threshold is studied, which makes it possible to compare the contractivity behavior of methods with an orderp>1 as well.
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.
TL;DR: In this article, a path-integral solution for processes described by nonlinear Fokker-Planck equations together with externally imposed boundary conditions is derived, written in the form of a path sum for small time steps and contains a conventional volume integral, a surface integral which incorporates the boundary conditions.
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.
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.
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.
TL;DR: An optical matrix multiplier using two linear modulating arrays in which the columns of the first matrix to be multiplied control the modulation of one array and the rows of the second matrix control the other array is described in this paper.
Abstract: An optical matrix multiplier using two linear modulating arrays in which the columns of the first matrix to be multiplied control the modulation of one array and the rows of the second matrix control the other array. Light is directed through all combinations of elements on the two arrays and the resultant beams measured by individual elements on a two-dimensional detector array. The detecting elements time integrates the intensity of light falling on each of them, which value corresponds to an element of the product of the two matrices. The invention may be implemented among other ways with two linear electrooptical arrays, a linear array of light emitting diodes and a linear electrooptical array, or two Bragg cells and a pulsed light source.
TL;DR: In this article, the Finite Element Method (FEM) and the Method of Moments (MOM) were applied to a pair of coupled integral equations to solve the static conductor-dielectric problems.
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.
TL;DR: Garrett et al. as discussed by the authors used a new numerical method for solving Schrodinger's equation in Delves' coordinates and showed that 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 8, 300-2400 K, for all three reactions with very large inertial effects.
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
TL;DR: This chapter discusses A-stable Methods, which focuses on the application of the Jain-Kutta method to Linear Multistep Systems, and its application to Highly Oscillatory Systems.
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.
TL;DR: In this article, the numerical integration of partial differential equations of the advection-diffusion type was studied and it was shown that finite differencing of the total derivatives yields schemes which do not require upwinding.
TL;DR: In this paper, a bilinear isoparametric finite element concept is used for the numerical analysis of multilayered plates, which allows for transverse shear and normal strains in each layer, thus extending the analysis to very thick plates and laminates.
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