Multidomain Spectral Method Research Articles

Predefined Exponential Basis Set for Half-Bounded Multi Domain Spectral Method

A non-orthogonal predefined exponential basis set is used to handle half-bounded domains in multi domain spectral method (MDSM). This approach works extremely well for real-valued semi-infinite differential problems. It spans simultaneously wide range of exponential decay rates with multi scaling and does not suffer from zero crossing. These two conditions are necessary for many physical problems. For comparison, the method is used to solve different problems and compared with analytical and published results. The comparison exhibits the strengths and accuracy of the presented basis set.

Large eddy simulation of flow in an unbaffled stirred tank for different Reynolds numbers

Large eddy simulation of flow in a stirred tank has been carried out for three different Reynolds numbers of 4000, 16 000, and 64 000 using a spectral multidomain method in order to observe the effect of Reynolds numbers on the flow fields. As the Reynolds number increases, each large ring vortex at the upper and lower parts of the tank has been broken up into smaller vortices. With increasing Reynolds number, the ring vortex in the mean field becomes a slightly more oblate and increases in height until it reaches the lid of the tank. The isosurface of swirling strength has been adopted to identify the instantaneous and mean tip-vortex structures. Due to the time-dependent jitter in the position of tip vortices, the time-averaged vortex structure appears to decay much faster. This decay is more rapid with increasing Reynolds number. The increment of the ratio of time and volume-averaged resolved scale dissipation and subgrid scaled dissipation was governed by the linear function of Reynolds number.

Comparative analysis of spectral methods in half-bounded domains; implementation of predefined exponential and Laguerre basis sets to study planar dielectric and plasmonic waveguides

Two types of basis sets are used to analyze half bounded domains within the frame of multi-domain spectral method, namely the predefined exponential and physical Laguerre basis sets. Different planar waveguides are used for comparisons and the comparisons demonstrate the superiority of the predefined exponential basis set. The physical Laguerre basis set suffers from the slow convergence and its high sensitivity to scaling. Also, a narrow range of exponentially decaying rates can be calculated simultaneously. On the other hand, the predefined exponential basis set can handle wide spectrum of decaying range simultaneously with a small number of bases. It results in accuracies comparable to the machine accuracy with about 20 bases in most of the presented analyses.

Keplerian frequency of uniformly rotating neutron stars and strange stars

<i>Aims. <i/>We calculate Keplerian (mass shedding) configurations of rigidly rotating neutron stars and strange stars with crusts. We check the validity of the empirical formula for Keplerian frequency, <i>f<i/><sub>K<sub/>, proposed by Lattimer & Prakash, , where <i>M<i/> is the (gravitational) mass of the Keplerian configuration, <i>R<i/> is the (circumferential) radius of the non-rotating configuration of the same gravitational mass, and kHz.<i>Methods. <i/>Numerical calculations are performed using precise 2D codes based on the multi-domain spectral methods. We use a representative set of equations of state (EOSs) of neutron stars and quark stars.<i>Results. <i/>We show that the empirical formula for holds within a few percent for neutron stars with realistic EOSs, provided , where is the maximum allowable mass of non-rotating neutron stars for an EOS, and kHz. Similar precision is obtained for strange stars with . For maximal crust masses we obtain kHz, and the value of <i>C<i/><sub>SS<sub/> is not very sensitive to the crust mass. All our <i>C<i/>s are significantly larger than the analytic value from the relativistic Roche model, kHz. For , the equatorial radius of the Keplerian configuration of mass <i>M<i/>, , is, to a very good approximation, proportional to the radius of the non-rotating star of the same mass, , with . The value of <i>a<i/><sub>SS<sub/> is very weakly dependent on the mass of the crust of the strange star. Both <i>a<i/> values are smaller than the analytic value from the relativistic Roche model.

Multi-domain spectral method for modal analysis of optical waveguide

A multi-domain spectral method for scalar optical modal analysis is developed and presented. The method is mesh-free and is applicable to three-dimensional optical waveguide structures that are invariant in one direction. The mode profile is represented as a superposition of one-dimensional preselected basis functions with variable coefficients. As common to spectral methods, the presented method is inherently fast and efficient. Different optical waveguides are analyzed for comparison purpose to exhibit the validity and accuracy of the presented method.

IMEX Evolution of Scalar Fields on Curved Backgrounds

Inspiral of binary black holes occurs over a time-scale of many orbits, far longer than the dynamical time-scale of the individual black holes. Explicit evolutions of a binary system therefore require excessively many time steps to capture interesting dynamics. We present a strategy to overcome the Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern implicit-explicit ODE solvers and multidomain spectral methods for elliptic equations. Our analysis considers the model problem of a forced scalar field propagating on a generic curved background. Nevertheless, we encounter and address a number of issues pertinent to the binary black hole problem in full general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild coordinates, we document the results of several numerical experiments testing our strategy.

Semiconductor nanodevice simulation by multidomain spectral method with Chebyshev, prolate spheroidal and Laguerre basis functions

A new approach based on spectral method with efficient basis functions is proposed in the paper to simulate semiconductor nanodevice by solving the Schrödinger equation. The computational domain is partitioned at heterojunctions into a number of subdomains. The envelope functions in subdomains are expanded by various efficient basis functions and then patched by the BenDaniel–Duke boundary conditions to preserve exponential order of accuracy. Importantly, the consideration to choose the basis functions depends on the oscillatory characteristics of envelope functions. Three kinds of basis functions including prolate spheroidal wave functions, Chebyshev polynomials, and Laguerre–Gaussian functions are used according to the mathematical features in this work. In addition, the determinations of optimum values of scaling factor in Laguerre–Gaussian functions and bandwidth parameter in prolate spheroidal wave functions are also discussed in detail. Several quantum well examples are simulated to validate the effectiveness of the present scheme. The relative errors of energy levels achieve the order of 10 −12 requiring merely a few grid points.

Linear perturbation response of self-similar ablative flows relevant to inertial confinement fusion

A family of exact similarity solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction is proposed for studying unsteadiness and compressibility effects on the hydrodynamic stability of ablation fronts relevant to inertial confinement fusion. Dynamical multi-domain Chebyshev spectral methods are employed for computing both the similarity solution and its time-dependent linear perturbations. This approach has been exploited to analyse the linear stability properties of two self-similar ablative configurations subjected to direct laser illumination asymmetries. Linear perturbation temporal and reduced responses are analysed, evidencing a maximum instability for illumination asymmetries of zero transverse wavenumber as well as three distinct regimes of ablation-front distortion evolution, and emphasizing the importance of the mean flow unsteadiness, compressibility and stratification.

Modelling of nonlinear dynamics for reciprocal multi-layer piezoceramic transducer systems

The dynamics of multi-layer transducer systems are modelled and simulated by use of a multi-domain spectral method. The model accounts for a nonlinear constitutive relation between the electric displacement and the electric field. Simulation of resonance and wave propagation in the transducer is implemented using a domain-decomposition method and governing equations in each layer are discretized using a Chebyshev collocation method. The model is applied to a reciprocal transducer system, as used in many engineering applications including ultrasonic flowmeter applications and blood-velocity measurement setups, where the transducer consists of a PZT5 piezoceramic layer and a matching layer.

Effective numerical viscosity in spectral multidomain penalty method-based simulations of localized turbulence

The numerical dissipation operating in a specific spectral multidomain method model developed for the simulation of incompressible high Reynolds number turbulence in doubly periodic domains is investigated. The method employs Fourier discretization in the horizontal directions and the discretization in the vertical direction is based on a Legendre collocation scheme local to each subdomain. Both spatial discretizations are characterized by either no or near-negligible artificial dissipation. In high Reynolds number simulations, which are inherently under-resolved, stability of the numerical scheme is ensured through spectral filtering in all three directions and the implementation of a penalty scheme in the vertical direction. The dissipative effects of these stabilizers are quantified in terms of the numerical viscosity, using a generalization of the method previously employed to analyze numerical codes for the simulation of homogeneous, isotropic turbulence in triply periodic domains. Data from simulations of the turbulent wake of a towed sphere are examined at two different Reynolds numbers varying by a factor of twenty. The effects of the stabilizers are found to be significant, i.e. comparable, and sometimes larger, than the effects of the physical (molecular) viscosity. Away from subdomain interfaces, the stabilizers have an expected dissipative character extending over a range of scales determined by timestep and the degree of under-resolution, i.e. Reynolds number. At the interfaces, the stabilizers tend to exhibit a strong anti-dissipative character. Such behavior is attributed to the inherently discontinuous formulation of the penalty scheme, which suppresses catastrophic Gibbs oscillations by enforcing C 0 and C 1 continuity only weakly at the interfaces.

The maximum mass of differentially rotating neutron stars

Binary neutron star merger lead to the formation of a massive, rapidly and differentially rotating neutron star (or a strange star) or to the prompt collapse to a black hole. The maximum mass of a differentially rotating remnant is crucial for distinguishing between these two final objects. We study the effect of differential rotation on the maximum mass of neutron stars (strange stars). We numerically construct stellar models using a highly accurate relativistic code based on a multi-domain spectral method. We find much larger mass increases for strange stars than for neutron stars for the same degree of differential rotation.

Aerodynamic and aeroacoustic analysis of a pulsating internal flow by a multidomain weak collocation spectral method

A numerical methodology is presented for the coupled aerodynamic and aeroacoustic analysis of time dependent laminar viscous flows in general two-dimensional geometries. The overall procedure is constituted firstly by the solution of the incompressible Navier–Stokes equations and secondly by the solution of a linear evolution equation of second order in space and time which originates from Lighthill's acoustic analogy. A semi-implicit projection method is utilized for the time integration of the incompressible Navier–Stokes system, while a choice between a second order implicit Newmark scheme and a fourth order explicit Runge–Kutta–Nyström method is offered for the temporal discretization of the wave equation. The multidomain weak Legendre collocation spectral method on quadrilateral subdomain topologies is employed for the spatial approximation of both the fluid dynamic and the acoustic problems. A specific pulsating internal flow inside a plane constricted channel is selected as a representative application for the assessment of the capabilities of the proposed discrete algorithm. Numerous results are presented and discussed, so as to thoroughly demonstrate the behavior of the numerical procedure.

Multidomain spectral method for the helically reduced wave equation

We consider the 2+1 and 3+1 scalar wave equations reduced via a helical Killing field, respectively referred to as the 2-dimensional and 3-dimensional helically reduced wave equation (HRWE). The HRWE serves as the fundamental model for the mixed-type PDE arising in the periodic standing wave (PSW) approximation to binary inspiral. We present a method for solving the equation based on domain decomposition and spectral approximation. Beyond describing such a numerical method for solving strictly linear HRWE, we also present results for a nonlinear scalar model of binary inspiral. The PSW approximation has already been theoretically and numerically studied in the context of the post-Minkowskian gravitational field, with numerical simulations carried out via the “eigenspectral method.” Despite its name, the eigenspectral technique does feature a finite-difference component, and is lower-order accurate. We intend to apply the numerical method described here to the theoretically well-developed post-Minkowski PSW formalism with the twin goals of spectral accuracy and the coordinate flexibility afforded by global spectral interpolation.

Erratum: Accurate and realistic initial data for black hole-neutron star binaries [Phys. Rev. D74, 124002 (2006)

This paper is devoted to the computation of compact binaries composed of one black hole and one neutron star. The objects are assumed to be on exact circular orbits. Standard 3+1 decomposition of Einstein equations is performed and the conformal flatness approximation is used. The obtained system of elliptic equations is solved by means of multi-domain spectral methods. Results are compared with previous work both in the high mass ratio limit and for one neutron star with very low compactness parameter. The accuracy of the present code is shown to be greater than with previous codes. Moreover, for the first time, some sequences containing one neutron star of realistic compactness are presented and discussed.

A multi-domain spectral method for initial data of arbitrary binaries in general relativity

We present a multi-domain spectral method to compute initial data of binary systems in general relativity. By utilizing adapted conformal coordinates, the vacuum region exterior to the gravitational sources is divided into two subdomains within which the spectral expansion of the field quantities is carried out. If a component of the binary is a neutron star, a further subdomain covering the star's interior is added. As such, the method can be used to construct arbitrary initial data corresponding to binary black holes, binary neutron stars or mixed binaries. In particular, it is possible to describe a black hole component by the puncture ansatz as well as through an excision technique. First examples are given for binary black hole excision data that fulfil the requirements of the quasi-stationary framework, which combines the conformal thin sandwich formulation of the constraint equations with the isolated horizon conditions for black holes in quasi-equilibrium. These numerical solutions were obtained to extremely high accuracy with moderate computational effort. Moreover, the method proves to be applicable even when tending towards limiting cases such as large mass ratios of the binary components.

Subsonic compressible flow in two-sided lid-driven cavity. Part I: Equal walls temperatures

This paper presents a numerical study of the laminar, viscous, subsonic compressible flow in a two-dimensional, two-sided, lid-driven cavity using a multi-domain spectral element method. The flow is driven by steadily moving two opposite walls vertically in opposite directions. All the bounding walls have equal temperatures. The results of the simulations are used to investigate the effects of the cavity aspect ratio, the Reynolds number and the Mach number on the flow. At lower Reynolds numbers, the flow pattern consists of two separate co-rotating vortices contiguous to the moving walls. For higher Reynolds numbers, initially a two-vortex flow is formed, which eventually turns into a single elliptical vortex occupying most of the cavity. For a higher aspect ratio, the flow patterns are dissimilar in that the streamlines become more and more elliptic. For aspect ratios as high as 2.5, at high Reynolds numbers, a three-vortex stage is formed. It is found that the compressibility effects are not very significant for Mach numbers less than 0.4. Dissipation of kinetic energy into internal energy changes the temperature field especially near the boundaries. Boundary layer studies suggest that the velocity and temperature boundary layer thicknesses are lower for higher Reynolds numbers. For engineering purposes, these thicknesses can be approximated by the existing flat-plate solutions.

Accurate and realistic initial data for black hole–neutron star binaries

This paper is devoted to the computation of compact binaries composed of one black hole and one neutron star. The objects are assumed to be on exact circular orbits. Standard $3+1$ decomposition of Einstein equations is performed and the conformal flatness approximation is used. The obtained system of elliptic equations is solved by means of multidomain spectral methods. Results are compared with previous work both in the high mass ratio limit and for one neutron star with very low compactness parameter. The accuracy of the present code is shown to be greater than with previous codes. Moreover, for the first time, some sequences containing one neutron star of realistic compactness are presented and discussed.

The final phase of inspiral of neutron stars: Realistic equations of state

Coalescing compact star binaries are expected to be among the strongest sources of gravitational radiation to be seen by laser interferometers. We present calculations of the final phase of inspiral of equal mass irrotational neutron star binaries and strange quark star binaries. Six types of equations of state at zero temperature are used – three realistic nuclear equations of state of various softness and three different MIT bag models of strange quark matter. We study the precoalescing stage within the Isenberg–Wilson–Mathews approximation of general relativity using a multidomain spectral method. The gravitational-radiation driven evolution of the binary system is approximated by a sequence of quasi-equilibrium configurations at fixed baryon number and decreasing separation. We find that the innermost stable circular orbit (ISCO) is given by an orbital instability for binary strange quark stars and by the mass-shedding limit for neutron star binaries. The gravitational wave frequency at the ISCO, which marks the end of the inspiral phase, is found to be ∼1100–1460 Hz for two 1.35 M ⊙ irrotational strange stars described by the MIT bag model and between 800 and 1230 Hz for neutron stars.

원형 핀이 부착된 실린더 주위의 유동 및 열전달에 관한 수치적 연구

Three-dimensional and time-dependent solution for the fluid flow and heat transfer past a circular cylinder with fins is obtained using accurate and efficient spectral methods. A Fourier expansion with a corresponding uniform grid is used along the circumferential direction. A spectral multi-domain method with a corresponding Chebyshev collocation is used along r-z plane to handle fins attached to the surface of a circular cylinder. At the Reynolds number of 300 based on a cylinder diameter, results with fins are compared with those without fins in order to see the effects of the presence of fins on three-dimensional and unsteady fluid flow and heat transfer past a bluff body. The detail structures of fluid flow and temperature field are obtained as a function of time to investigate how the presence of fins changes heat transfer mechanism related to the vortical structure in the wake region.

A Conservative Isothermal Wall Boundary Condition for the Compressible Navier–Stokes Equations

We present a conservative isothermal wall boundary condition treatment for the compressible Navier-Stokes equations. The treatment is based on a manipulation of the Osher solver to predict the pressure and density at the wall, while specifying a zero boundary flux and a fixed temperature. With other solvers, a non-zero mass flux occurs through a wall boundary, which is significant at low resolutions in closed geometries. A simulation of a lid driven cavity flow with a multidomain spectral method illustrates the effect of the new boundary condition treatment.