Results for “spectral methods” — Page 7

NASA NTRS1981

Spectral multigrid methods for elliptic equations

An alternative approach which employs multigrid concepts in the iterative solution of spectral equations was examined. Spectral multigrid methods are described for self adjoint elliptic equations with either periodic or Dirichlet boundary conditions. For realistic fluid calculations the relevant boundary conditions are periodic in at least one (angular) coordinate and Dirichlet (or Neumann) in the remaining coordinates. Spectral methods are always effective for flows in strictly rectangular geometries since corners generally introduce singularities into the solution. If the boundary is smooth, then mapping techniques are used to transform the problem into one with a combination of periodic and Dirichlet boundary conditions. It is suggested that spectral multigrid methods in these geometries can be devised by combining the techniques.

Zang, T. A.↗

NASA NTRS1983

Spectral multigrid methods with applications to transonic potential flow

Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil.

Streett, C. L.↗

NASA NTRS1985

Spectral multigrid methods with applications to transonic potential flow

Spectral multigrid methods are demonstrated to be a competitive technique for solving the transonic potential flow equation. The spectral discretization, the relaxation scheme, and the multigrid techniques are described in detail. Significant departures from current approaches are first illustrated on several linear problems. The principal applications and examples, however, are for compressible potential flow. These examples include the relatively challenging case of supercritical flow over a lifting airfoil.

Streett, C. L.↗

NASA NTRS1982

Shock-fitted Euler solutions to shock vortex interactions

The interaction of a planar shock wave with one or more vortexes is computed using a pseudospectral method and a finite difference method. The development of the spectral method is emphasized. In both methods the shock wave is fitted as a boundary of the computational domain. The results show good agreement between both computational methods. The spectral method is, however, restricted to smaller time steps and requires use of filtering techniques.

Salas, M. D.↗

NASA NTRS1982

Shock-fitted Euler solutions to shock-vortex interactions

The interaction of a planar shock wave with one or more vortexes is computed using a pseudospectral method and a finite difference method. The development of the spectral method is emphasized. In both methods the shock wave is fitted as a boundary of the computational domain. The results show good agreement between both computational methods. The spectral method is, however, restricted to smaller time steps and requires use of filtering techniques. Previously announced in STAR as N82-28061

Salas, M. D.↗

NASA NTRS2002

High-Order Spectral Volume Method for 2D Euler Equations

The Spectral Volume (SV) method is extended to the 2D Euler equations. The focus of this paper is to study the performance of the SV method on multidimensional non-linear systems. Implementation details including total variation diminishing (TVD) and total variation bounded (TVB) limiters are presented. Solutions with both smooth features and discontinuities are utilized to demonstrate the overall capability of the SV method.

Wang, Z. J.↗

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.↗

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 NTRS1985

On spectral multigrid methods for the time-dependent Navier-Stokes equations

A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.

Zang, T. A.↗

NASA NTRS1986

On spectral multigrid methods for the time-dependent Navier-Stokes equations

A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.

Zang, T. A.↗

NASA NTRS1986

An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry

High-order (p-type) finite element methods combine the geometric flexibility of standard low-order finite element (or finite volume) techniques with the rapid convergence properties of spectral methods. Various p-type schemes have been proposed for elliptic problems. However, an application of p-type methods to more complex equations, in particular the passive scalar or Navier-Stokes equations, is complicated by the nature of the equations and by efficiency considerations as regards the solution procedure. Patera (1984) has proposed a p-type method for the Navier-Stokes equations, taking into account the arising additional complications. The present study is concerned with an isoparametric spectral element discretization for the solution of flow problems in fully general, curvy geometries. Attention is given to the isoparametric formulation for elliptic equations and a time-splitting scheme for the Navier-Stokes equations.

Korczak, K. Z.↗

NASA NTRS1984

A spectral element method for fluid dynamics - Laminar flow in a channel expansion

A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work.

Patera, A. T.↗

NASA NTRS1989

Essentially nonoscillatory spectral Fourier methods for shock wave calculations

An essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations is presented. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution.

Cai, Wei↗

NASA NTRS1988

Non-oscillatory spectral Fourier methods for shock wave calculations

A non-oscillatory spectral Fourier method is presented for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.

Cai, Wei↗

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 NTRS1986

A spectral element method for the simulation of unsteady incompressible flows with heat transfer

The spectral element method is a high-order finite element technique for solution of the Navier-Stokes and energy equations. In the isoparametric spectral element discretization, the domain is broken up into general brick elements, and the dependent and independent variables represented as high-order tensor-product Lagrangian interpolants through Chebyshev collocation points. The nonlinear and convective terms in the governing equations are treated with explicit collocation, while the pressure and diffusive contributions are handled implicitly using variational projection operators. The method is applied to flow past a cylinder, flow in grooved channels, and natural convection in an enclosure.

Karniadakis, George E.↗

NASA NTRS1998

Multi-Dimensional High Order Essentially Non-Oscillatory Finite Difference Methods in Generalized Coordinates

This project is about the development of high order, non-oscillatory type schemes for computational fluid dynamics. Algorithm analysis, implementation, and applications are performed. Collaborations with NASA scientists have been carried out to ensure that the research is relevant to NASA objectives. The combination of ENO finite difference method with spectral method in two space dimension is considered, jointly with Cai [3]. The resulting scheme behaves nicely for the two dimensional test problems with or without shocks. Jointly with Cai and Gottlieb, we have also considered one-sided filters for spectral approximations to discontinuous functions [2]. We proved theoretically the existence of filters to recover spectral accuracy up to the discontinuity. We also constructed such filters for practical calculations.

Shu, Chi-Wang↗