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Convergence acceleration for vector sequences and applications to computational fluid dynamics
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
Convergence acceleration for vector sequences and applications to computational fluid dynamics
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
Convergence acceleration for least squares differential corrections
Convergence acceleration for least squares differential corrections
Convergence acceleration of Monte Carlo many-body perturbation methods by using many control variates
Herein, the use of many control variates is proposed as a method to accelerate the second- and third-order Monte Carlo (MC) many-body perturbation (MC-MP2 and MC-MP3) calculations. A control variate is an exactly integrable function that is strongly correlated or anti-correlated with the target function to be integrated by the MC method. Evaluating both integrals as well as their covariances in the same MC run, one can effect a mutual cancellation of the statistical uncertainties and biases in the MC integrations, thereby accelerating its convergence considerably. Six and thirty-six control variates, whose integrals are known a priori, are generated for MC-MP2 and MC-MP3, respectively, by systematically replacing one or more two-electron-integral vertexes of certain configurations by zero-valued overlap-integral vertexes in their Goldstone diagrams. The variances and co- variances of these control variates are computed at a marginal cost, enhancing the overall efficiency of the MC-MP2 and MC-MP3 calculations by a factor of up to 14 and 20, respectively.
A stochastic approximation scheme with accelerated convergence properties
Stochastic approximation scheme with accelerated convergence properties
Non-LTE radiative transfer with lambda-acceleration - Convergence properties using exact full and diagonal lambda-operators
We investigate the convergence properties of Lambda-acceleration methods for non-LTE radiative transfer problems in planar and spherical geometry. Matrix elements of the 'exact' A-operator are used to accelerate convergence to a solution in which both the radiative transfer and atomic rate equations are simultaneously satisfied. Convergence properties of two-level and multilevel atomic systems are investigated for methods using: (1) the complete Lambda-operator, and (2) the diagonal of the Lambda-operator. We find that the convergence properties for the method utilizing the complete Lambda-operator are significantly better than those of the diagonal Lambda-operator method, often reducing the number of iterations needed for convergence by a factor of between two and seven. However, the overall computational time required for large scale calculations - that is, those with many atomic levels and spatial zones - is typically a factor of a few larger for the complete Lambda-operator method, suggesting that the approach should be best applied to problems in which convergence is especially difficult.
Convergence acceleration procedure for the method of steepest descent
Convergence acceleration procedure for steepest descent method for trajectory optimization
On Convergence Acceleration Techniques for Unstructured Meshes
A discussion of convergence acceleration techniques as they relate to computational fluid dynamics problems on unstructured meshes is given. Rather than providing a detailed description of particular methods, the various different building blocks of current solution techniques are discussed and examples of solution strategies using one or several of these ideas are given. Issues relating to unstructured grid CFD problems are given additional consideration, including suitability of algorithms to current hardware trends, memory and cpu tradeoffs, treatment of non-linearities, and the development of efficient strategies for handling anisotropy-induced stiffness. The outlook for future potential improvements is also discussed.
Convergence acceleration of viscous flow computations
A multiple-grid convergence acceleration technique introduced for application to the solution of the Euler equations by means of Lax-Wendroff algorithms is extended to treat compressible viscous flow. Computational results are presented for the solution of the thin-layer version of the Navier-Stokes equations using the explicit MacCormack algorithm, accelerated by a convective coarse-grid scheme. Extensions and generalizations are mentioned.
Numerical comparisons of nonlinear convergence accelerators
As part of a continuing program of numerical tests of convergence accelerators, the iterated Aitken's Delta-squared method, Wynn's epsilon algorithm, Brezinski's theta algorithm, and Levin's u transform are compared on a broad range of test problems: linearly convergence alternating, monotone, and irregular-sign series, logarithmically convergent series, power method and Bernoulli method sequences, alternating and monotone asymptotic series, and some perturbation series arising in applications. In each category either the epsilon algorithm or the u transform gives the best results of the four methods tested. In some cases differences among methods are slight, and in others they are quite striking.
On the construction of solutions of linear differential equations with quasiperiodical coefficients using the method of accelerated convergence
Linear differential equations with quasi- periodical coefficients solved using method of accelerated convergence
Accelerating convergence of one-step methods for the numerical solution of ordinary differential equations.
Accelerating convergence of one step methods for numerical solution of ordinary differential equations, discussing various initial value problems
Convergence acceleration of the Proteus computer code with multigrid methods
This report presents the results of a study to implement convergence acceleration techniques based on the multigrid concept in the two-dimensional and three-dimensional versions of the Proteus computer code. The first section presents a review of the relevant literature on the implementation of the multigrid methods in computer codes for compressible flow analysis. The next two sections present detailed stability analysis of numerical schemes for solving the Euler and Navier-Stokes equations, based on conventional von Neumann analysis and the bi-grid analysis, respectively. The next section presents details of the computational method used in the Proteus computer code. Finally, the multigrid implementation and applications to several two-dimensional and three-dimensional test problems are presented. The results of the present study show that the multigrid method always leads to a reduction in the number of iterations (or time steps) required for convergence. However, there is an overhead associated with the use of multigrid acceleration. The overhead is higher in 2-D problems than in 3-D problems, thus overall multigrid savings in CPU time are in general better in the latter. Savings of about 40-50 percent are typical in 3-D problems, but they are about 20-30 percent in large 2-D problems. The present multigrid method is applicable to steady-state problems and is therefore ineffective in problems with inherently unstable solutions.
Convergence acceleration of the Proteus computer code with multigrid methods
Presented here is the first part of a study to implement convergence acceleration techniques based on the multigrid concept in the Proteus computer code. A review is given of previous studies on the implementation of multigrid methods in computer codes for compressible flow analysis. Also presented is a detailed stability analysis of upwind and central-difference based numerical schemes for solving the Euler and Navier-Stokes equations. Results are given of a convergence study of the Proteus code on computational grids of different sizes. The results presented here form the foundation for the implementation of multigrid methods in the Proteus code.
Convergence acceleration of viscous and inviscid hypersonic flow calculations
The convergence of inviscid and viscous hypersonic flow calculations using a two-dimensional flux-splitting code is accelerated by applying a Richardson-type overrelaxation method. Successful results are presented for various cases; and a 50 percent savings in computer time is usually achieved. An analytical formula for the overrelaxation factor is derived, and the performance of this scheme is confirmed numerically. Moreover, application of this overrelaxation scheme produces a favorable preconditioning for Wynn's epsilon-algorithm. Both techniques have been extended to viscous three-dimensional flows and applied to accelerate the convergence of the compressible Navier-Stokes code. A savings of 40 percent in computer time is achieved in this case.
Reliability enhancement of Navier-Stokes codes through convergence acceleration
Methods for enhancing the reliability of Navier-Stokes computer codes through improving convergence characteristics are presented. The improving of these characteristics decreases the likelihood of code unreliability and user interventions in a design environment. The problem referred to as a 'stiffness' in the governing equations for propulsion-related flowfields is investigated, particularly in regard to common sources of equation stiffness that lead to convergence degradation of CFD algorithms. Von Neumann stability theory is employed as a tool to study the convergence difficulties involved. Based on the stability results, improved algorithms are devised to ensure efficient convergence in different situations. A number of test cases are considered to confirm a correlation between stability theory and numerical convergence. The examples of turbulent and reacting flow are presented, and a generalized form of the preconditioning matrix is derived to handle these problems, i.e., the problems involving additional differential equations for describing the transport of turbulent kinetic energy, dissipation rate and chemical species. Algorithms for unsteady computations are considered. The extension of the preconditioning techniques and algorithms derived for Navier-Stokes computations to three-dimensional flow problems is discussed. New methods to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equtions are developed, with a special emphasis on the acceleration of convergence on highly clustered grids.
Accelerated convergence for synchronous approximate agreement
The protocol for synchronous approximate agreement presented by Dolev et. al. exhibits the undesirable property that a faulty processor, by the dissemination of a value arbitrarily far removed from the values held by good processors, may delay the termination of the protocol by an arbitrary amount of time. Such behavior is clearly undesirable in a fault tolerant dynamic system subject to hard real-time constraints. A mechanism is presented by which editing data suspected of being from Byzantine-failed processors can lead to quicker, predictable, convergence to an agreement value. Under specific assumptions about the nature of values transmitted by failed processors relative to those transmitted by good processors, a Monte Carlo simulation is presented whose qualitative results illustrate the trade-off between accelerated convergence and the accuracy of the value agreed upon.