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Results for “spectral collocation methods”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

A Chebyshev spectral collocation method using a staggered grid for the stability of cylindrical flows

A staggered spectral collocation method for the stability of cylindrical flows is developed. In this method the pressure is evaluated at different nodal points than the three velocity components. These modified nodal points do not include the two boundary nodes; therefore the need for the two artificial pressure boundary conditions employed by Khorrami et al. is eliminated. It is shown that the method produces very accurate results and has a better convergence rate than the spectral tau formulation. However, through extensive convergence tests it was found that elimination of the artificial pressure boundary conditions does not result in any significant change in the convergence behavior of spectral collocation methods.

Khorrami, Mehdi R.↗
NASA NTRS1991

A spectral collocation method for compressible, non-similar boundary layers

An efficient and highly accurate algorithm based on a spectral collocation method is developed for numerical solution of the compressible, two-dimensional and axisymmetric boundary layer equations. The numerical method incorporates a fifth-order, fully implicit marching scheme in the streamwise (timelike) dimension and a spectral collocation method based on Chebyshev polynomial expansions in the wall-normal (spacelike) dimension. The spectral collocation algorithm is used to derive the nonsimilar mean velocity and temperature profiles in the boundary layer of a 'fuselage' (cylinder) in a high-speed (Mach 5) flow parallel to its axis. The stability of the flow is shown to be sensitive to the gradual streamwise evolution of the mean flow and it is concluded that the effects of transverse curvature on stability should not be ignored routinely.

Pruett, C. D.↗
NASA NTRS1987

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2.

Hussaini, M. Y.↗
NASA NTRS1989

Spectral collocation methods

This review covers the theory and application of spectral collocation methods. Section 1 describes the fundamentals, and summarizes results pertaining to spectral approximations of functions. Some stability and convergence results are presented for simple elliptic, parabolic, and hyperbolic equations. Applications of these methods to fluid dynamics problems are discussed in Section 2.

Hussaini, M. Y.↗
DOE OSTI2024

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Domain decomposition preconditioners for the spectral collocation method

Several block iteration preconditioners are proposed and analyzed for the solution of elliptic problems by spectral collocation methods in a region partitioned into several rectangles. It is shown that convergence is achieved with a rate which does not depend on the polynomial degree of the spectral solution. The iterative methods here presented can be effectively implemented on multiprocessor systems due to their high degree of parallelism.

Quarteroni, Alfio↗
NASA NTRS1989

A multidomain spectral collocation method for the Stokes problem

A multidomain spectral collocation scheme is proposed for the approximation of the two-dimensional Stokes problem. It is shown that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.

Landriani, G. Sacchi↗
NASA NTRS1989

A pseudo-spectral collocation method applied to the problem of convective diffusive transport in fluids subject to unsteady residual accelerations

The problem of determining the sensitivity of Bridgman-Stockbarger directional solidification experiments to residual accelerations of the type associated with spacecraft in low earth orbit is analyzed numerically using a pseudo-spectral collocation method. The approach employs a novel iterative scheme combining the method of artificial compressibility and a generalized ADI method. The results emphasize the importance of the consideration of residual accelerations and careful selection of the operating conditions in order to take full advantages of the low gravity conditions.

Alexander, J. Iwan↗
NASA NTRS1984

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1985

A spectral collocation method for the Navier-Stokes equations

A Fourier-Chebyshev spectral method for the incompressible Navier-Stokes equations is described. It is applicable to a variety of problems including some with fluid properties which vary strongly both in the normal direction and in time. In this fully spectral algorithm, a preconditioned iterative technique is used for solving the implicit equations arising from semi-implicit treatment of pressure, mean advection and vertical diffusion terms. The algorithm is tested by applying it to hydrodynamic stability problems in channel flow and in external boundary layers with both constant and variable viscosity.

Malik, M. R.↗
NASA NTRS1990

Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel

The numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods is described. The key features of the method are the decomposition of the flow region into a number of rectangular subregions and spectral approximations which are pointwise C(1) continuous across subregion interfaces. Spectral approximations to the solution are obtained for Reynolds numbers in the range 0 to 500. The size of the salient corner vortex decreases as the Reynolds number increases from 0 to around 45. As the Reynolds number is increased further the vortex grows slowly. A vortex is detected downstream of the contraction at a Reynolds number of around 175 that continues to grow as the Reynolds number is increased further.

Karageorghis, Andreas↗
NASA NTRS1992

Spurious Modes in Spectral Collocation Methods with Two Non-Periodic Directions

Collocation implementation of the Kleiser-Schumann's method in geometries with two non-periodic directions is shown to suffer from three spurious modes - line, column and checkerboard - contaminating the computed pressure field. The corner spurious modes are also present but they do not affect evaluation of pressure related quantities. A simple methodology in the inversion of the influence matrix will efficiently filter out these spurious modes.

Balachandar, S.↗
NASA NTRS1991

Stability of compressible Taylor-Couette flow

Compressible stability equations are solved using the spectral collocation method in an attempt to study the effects of temperature difference and compressibility on the stability of Taylor-Couette flow. It is found that the Chebyshev collocation spectral method yields highly accurate results using fewer grid points for solving stability problems. Comparisons are made between the result obtained by assuming small Mach number with a uniform temperature distribution and that based on fully incompressible analysis.

Kao, Kai-Hsiung↗
NASA NTRS1984

Spectral methods for the Euler equations: Fourier methods and shock-capturing

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of Fourier spectral collocation methods are provided. Filtering strategies for shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one dimensional, periodic astrophysical ""nozzle'' problem.

Hussaini, M. Y.↗
NASA NTRS1983

Spectral methods for the Euler equations

Spectral methods for compressible flows are introduced in relation to finite difference and finite element techniques within the framework of the method of weighted residuals. Current spectral collocation methods are put in historical context. The basic concepts of both Fourier and Chebyshev spectral collocation methods are provided. Filtering strategies for both shock-fitting and shock-capturing approaches are also presented. Fourier shock capturing techniques are evaluated using a one-dimensional, periodic astrophysical 'nozzle' problem. Examples of shock-fitting approaches include a shock/acoustic wave interaction, shock/vortex interaction, and the classical blunt body problem. While the shock capturing spectral method does not yet show a clear advantage over second-order finite differences, equivalent accuracy can be obtained using shock fitting with far fewer grid points.

Hussaini, M. Y.↗
NASA NTRS2013

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations.

Carpenter, Mark H.↗
NASA NTRS1989

Domain decomposition methods for systems of conservation laws: Spectral collocation approximations

Hyperbolic systems of conversation laws are considered which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level a domain deposition method based on an iteration by subdomain procedure was introduced yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The method is set for a general nonlinear problem in several space variables. The convergence analysis, however, is carried out only for a linear one-dimensional system with continuous solutions. A precise form of the error reduction factor at each iteration is derived. Although the method is applied here to the case of spectral collocation approximation only, the idea is fairly general and can be used in a different context as well. For instance, its application to space discretization by finite differences is straight forward.

Quarteroni, Alfio↗