Implicit Operator Research Articles

Two-Dimensional Simulation of the Dynamics of the Collapse of a Rotating Core with Formation of a Neutron Star on an Adaptive Triangular Grid in Lagrangian Coordinates

Results are presented from a two-dimensional numerical simulation of the collapse of a rotating core with formation of a neutron star that has strong differential rotation in its outer regions. A specially developed numerical method is used which is based on a fully conservative implicit operator difference scheme for gravitational gas dynamics problems in lagrangian coordinates on a variable-structure triangular grid. The recoil shock wave generated by the collapse causes ejection of a small amount of material. This cannot explain the explosion of type II supernovae. The strong differential rotation in the presence of even a weak initial magnetic field obtained in these calculations must lead to a rise in the magnetic pressure, formation of an MHD shock wave, and conversion of rotational energy into the energy of radial expansion (magnetorotational supernova explosion).

Parallel Implementation of Curvilinear High-order Formulas

‘High-order formulas are required for differencing and filtering the Navier-Stokes equations in order to obtain the needed accuracy for a variety of CFD applications. The parallel performance issues relevant to one of these methods, the compact scheme, is studied -in this paper with emphasis on the associated implicit operators. Three procedures were selected: the one-sided method, the Parallel Diagonal Dominant method (PDD), and the Parallel Thomas Algorithm method (PTA). These parallel procedures were implemented in the AFRL code, FDLSDI. Kernel codes were also developed to extract some inherent performance features of these methods. Some of the calculations combine the methods for compact differencing and filtering. In general, the procedures based on the one-sided schemes produced accurate results and good parallel performance (efficiency, speedup, and scalability). The procedures that combine the one-sided schemes and PDD also performed well. However, parallel calculations that use the PDD method for both compact differencing and filtering produced the wrong results for loworder filters. On the other hand, high-order filters cause PDD to be very expensive. The one-sided method leads to superscalable calculations when the number of processors is few. For PDD, increasing the number of grid points in the derivative-difference direction leads to better speedup, as does.an increase in the number of right-hand side (FUXS) columns. In standard implementation (i.e., without engaO&g the processors during the idle time), the PTA procedure has a very poor parallel performance in comparison to PDD and the one-sided formulations. However, the procedure tends to be more accurate.

Numerical study of the global ocean equilibrium circulation

A new version of the global ocean general circulation model with σ -coordinates is presented. The modelled area approximates the whole World ocean basin from the coast of Antarctica to the North Pole. The model is designed for very long integration periods. Accordingly the numerical techniques applied are very efficient, using implicit timestepping and operator splitting methods. The model is driven at the surface by wind-stress and by restoring temperature and salinity to the observed annual cycle. The cyclo-stationary solution of the model is shown to depend on the initial conditions even after many thousand years of integration. Four different solutions are briefly reported in this paper. The existence of multiple equilibria has been reported previously only under mixed boundary conditions. The key mechanismof the multiple equilibria is nonlinear deep convection processes. Numerical experiments show that global mean values of temperature and salinity are rather important parameters affecting the quasi-equilibrium state.

Equational description of pseudovarieties of homomorphisms

The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in AC0 is given.

PROFINITE IDENTITIES FOR FINITE SEMIGROUPS WHOSE SUBGROUPS BELONG TO A GIVEN PSEUDOVARIETY

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the pseudovariety [Formula: see text] of all finite semigroups whose subgroups belong to a given finitely based pseudovariety H of finite groups.

Short Communication: A Parallel Newton–Krylov Method for Navier–Stokes Rotorcraft Codes

The application of Krylov subspace iterative methods to unsteady three-dimensional Navier–Stokes codes on massively parallel and distributed computing environments is investigated. Previously, the Euler mode of the Navier–Stokes flow solver Transonic Unsteady Rotor Navier–Stokes (TURNS) has been coupled with a Newton–Krylov scheme which uses two Conjugate-Gradient-like (CG) iterative methods. For the efficient implementation of Newton–Krylov methods to the Navier–Stokes mode of TURNS, efficient preconditioners must be used. Parallel implicit operators are used and compared as preconditioners. Results are presented for two-dimensional and three-dimensional viscous cases. The Message Passing Interface (MPI) protocol is used, because of its portability to various parallel architectures.

Tameness of Some Locally Trivial Pseudovarieties

Tameness is a strong property of semigroup pseudovarieties related to the membership problem. Let κ be the signature comprising semigroup multiplication and the omega implicit operation. We prove the κ-tameness of the pseudovarieties N, D, K and LI.

Numerical study of the global ocean equilibrium circulation

A new version of the global ocean general circulation model with σ -coordinates is presented. The modelled area approximates the whole World ocean basin from the coast of Antarctica to the North Pole. The model is designed for very long integration periods. Accordingly the numerical techniques applied are very efficient, using implicit timestepping and operator splitting methods. The model is driven at the surface by wind-stress and by restoring temperature and salinity to the observed annual cycle. The cyclo-stationary solution of the model is shown to depend on the initial conditions even after many thousand years of integration. Four different solutions are briefly reported in this paper. The existence of multiple equilibria has been reported previously only under mixed boundary conditions. The key mechanismof the multiple equilibria is nonlinear deep convection processes. Numerical experiments show that global mean values of temperature and salinity are rather important parameters affecting the quasi-equilibrium state.

Parallelization of Rotorcraft Aerodynamics Navier-Stokes Codes

The modie cation of unsteady three-dimensional Navier ‐Stokes codes for application on massively parallel and distributed computing environments is investigated. Previously, theEulermode of theNavier ‐Stokescode TURNS hasbeenparallelized. Fortheefe cientimplementation oftheNavier ‐StokesmodeofTURNS onparallelcomputing systems, several algorithmic changes should be made. The main modie cation is done on the implicit operator, lower‐upper symmetric Gauss ‐Seidel. Two new implicit operators are used because of convergence problems of traditional operators with high cell aspect ratio grids needed for viscous calculations. Results for Navier ‐Stokes cases are presented for various operators. The message passing interface protocol is used because of its portability to various parallel architectures. Nomenclature a = speed of sound e = total energy p = pressure Pr = Prandtl number Re = Reynolds number u;v;w = Cartesian velocities x; y;z = Cartesian coordinates ° = specie c heat ratio π = dynamic viscosity »;´;‡ = generalized coordinates Ω = density 9 = rotor azimuth angle

Syntax-semantics interface for tense and aspect in French

This paper proposes an HPSG account of the French tense and aspect system, focussing on the analysis of the passé simple (simple past) and imparfait (imperfective) tenses and their interaction with aspectually sensitive adjuncts. Starting from de Swart's (1998) analysis of the semantics of tense and aspect, I show that while the proposed semantic representations are appropriate, the analysis of implicit aspectual operators as coercion operators is inadequate. The proposed HPSG analysis relies on Minimal Recursion Semantics to relate standard syntactic structures with de Swart-style semantic representations. The analysis has two crucial features: first, it assumes that the semantic contribution of tense originates in the verb's semantic representation, despite the fact that tense can get wide scope over other semantic elements. Second, it allows the occurrence of implicit aspectual operators to be controlled by the verb's inflectional class, which accounts for their peculiar distribution.

Automata-Theoretic Decision Procedures for Information Logics

Automata-theoretic decision procedures for solving model-checking and satisfiability problems for temporal, dynamic, and description logics have flourished during the past decades. In the paper we define an EXPTIME decision procedure based on the emptiness problem of Buchi automata on infinite trees for the very expressive information logic SIM designed for reasoning about information systems. This logic involves modal parameters satisfying certain properties to capture the relevant nominals at the formula level, Boolean expressions and nominals at the modal level, an implicit intersection operation for relations, and a universal modality. The original combination of known techniques allows us to solve the open question related to the EXPTIME-completeness of SIM. Furthermore, we discuss how variants of SIM can be treated similarly although the decidability status of some of them is still unknown.

NORMAL FORMS FOR FREE APERIODIC SEMIGROUPS

The implicit operation ω is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using ω there is a well-defined algebra which is known as the free aperiodic semigroup. In this article we introduce a specific and rather elementary list of pseudoidentitites, we show that for each n, the n-generated free aperiodic semigroup is defined by this list of pseudoidentities, and then we use this identification to show that it has a decidable word problem. In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is κ-recursive. This completes a crucial step towards showing that the Krohn–Rhodes complexity of every finite semigroup is decidable.

Dynamics of implicit operations and tameness of pseudovarieties of groups

This work gives a new approach to the construction of implicit operations. By considering “higher-dimensional” spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesskiĭ, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of p p -groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups V \mathbf {V} , if it can be decided whether a finitely generated subgroup of the free group with the pro- V \mathbf {V} topology is dense, then V \mathbf {V} is tame.

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.

Linear-Quadratic Optimal Control Problems for Discrete Descriptor Systems

The control in the feedback form and the optimal cost are obtained for four linear-quadratic control problems by discrete descriptor systems in a Hilbert space. Special solutions of the implicit operator Riccati equations used here act in all state space and in general case they are not non-negative definite unlike the operators in the subspace and the non-negative definite solutions, acting in all state space, as it was in other authors papers. The examples show that the considered operator Riccati equations have no non-negative definite solutions though the corresponding optimal control problems have the unique solutions.

Free profinite locally idempotent and locally commutative semigroups

This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety L Sl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.

A specially combined lower–upper factored implicit scheme for three-dimensional compressible Navier–Stokes equations

A specially combined lower–upper factored implicit scheme based on the lower–upper symmetric-Gauss–Seidel (LU-SGS) implicit scheme in conjunction with diagonalization and Gaussian elimination (GE) is proposed for solving the compressible, three-dimensional Navier–Stokes equations and the two-equation q– ω turbulence model. The present scheme, LU-SGS-GE, contains all features of the LU-SGS scheme. Because the similarity transforms are used to construct upstream Jacobian matrices of flux vectors, it leads to much faster convergence and better stability without improper numerical dissipation and free-parameters; besides, the block-diagonal matrix inversions are still eliminated partly, and the implicit operator can also be vectorized completely. The new implicit scheme can be used for solving the unsteady three-dimensional flows.

On Algorithmic Problems for Joins of Pseudovarieties

This paper considers the algorithmic problems of decidability (of membership), decidability of pointlikes and idempotent pointlikes, and hyperdecidability for joins of pseudovarieties of semigroups. In particular, we show that if \pv V\subseteq \pv J has a decidable word problem for its semigroup of implicit operations and \pv W is a hyperdecidable pseudovariety of completely regular semigroups, then \pv V \vee \pv W is hyperdecidable. The same holds for pointlikes and idempotent pointlikes. In fact by careful analysis of the arguments involved, one can show that \pv J \vee \pv G is definable by a recursive set of pseudoidentities, involving only implicit operations built up from words and the ()ω -1 operator. Since Trotter and Volkov have proven that \pv J \vee \pv G is not finitely based, we feel that this is a satisfactory answer to the pseudoidentity basis problem for \pv J \vee \pv G , and in fact this basis gives an, albeit unbounded, membership algorithm.

Factorized Implicit Upwind Methods Applied to Inviscid Flows at High Mach Number

The performance of several approximate factorization methods, coupled with a e nite volume spatial discretization using Roe’ s approximate Riemann solver, are compared by means of numerical tests on a two-dimensional steady inviscid e ow past a blunt body at Mach numbers ranging from 5 to 20. The comparisons are carried out evaluating, by numerical experiments, the optimal Courant number of each method. The alternating direction implicit and the lower ‐upper symmetric Gauss ‐Seidel methods result in the most efe cient factorizations, in terms of CPU time. The former behaves smoothly with increasing Mach number, and its performance is not affected by the grid size. The latter may achieve higher efe ciency but is strongly dependent on the number of relaxation steps performed, requiring optimization in terms of Mach number and grid size. I. Introduction A N IMPLICIT time integration is usually required to improve convergence toward steady state in time-marching, shockcapturing methods. This requirement is particularly true for high Mach number real gas e ows where the practical stability limit of explicit methods is particularly severe. The large block banded linearized operators, resulting from the application of multistep implicitmethodsto multidimensionalstructuredgrids,aresolvedmost efe ciently by resorting to approximate factorization (AF). The implicit operator may be split along the grid directions, as in the early alternating direction implicit (ADI) schemes of Briley and McDonald 1 andBeamand Warming, 2 orintotwolower ‐upper(LU) factors, as in the original LU schemes of Steger and Warming 3 and Jameson and Turkel. 4 The ADI scheme, without the addition of an appropriate amount of numerical damping, may, however, become unstable in three dimensions when formulated in delta form. Therefore, the LU factorization has become increasingly popular, and many improvements to this technique have been proposed in recent years. 5 Yoon and Jameson, 6,7 combined the LU factoriza

Implicit High-Order-Accurate-in-Space Algorithms for the Navier-Stokes Equations

High-order-accurate, alternating direction implicit schemes have been developed. These schemes can be used to obtain efe ciently high-resolution numerical solutions of the compressible Euler and Navier ‐Stokes equations. Implicit solvers compatible with both centered and upwind schemes for the discretization of the convective terms arediscussed.Forcenteredspacediscretizationoftheconvectivee uxes,fourth-orderspatialaccuracyoftheimplicit operators is obtained at no additional computing cost by performing compact space differentiation. This implicit compact scheme, which requires block-tridiagonal matrix inversions, is used to obtain viscous and inviscid e ow solutions in curvilinear coordinates. In addition, high-order-accurate-in-space implicit operators are obtained for diagonalized schemes, which require scalar matrix inversions. Both high-order algorithms improve convergence characteristics for subsonic and transonic e ow compared to second-order-accurate-in-space implicit schemes. High-order-accurate-in-space implicit algorithms, compatible with upwind, high-resolution shock-capturing total variation diminishing schemes, are also developed. Third-order-accurate, upwind-biased, diagonalized algorithms were found to be as accurate and more efe cient than the standard block-tridiagonal, implicit solvers.