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Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations

Numerical Mathematics(Theory,Methods and Applications) 2011年第2期4卷 113-141页

作者： F.Auteri l.quartapelle Politecnico di Milano Dipartimento di Ingegneria AerospazialeVia La Masa 3420156 MilanoItaly

In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center,it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal *** idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical *** the Dirichlet boundary value problem in both geometries,original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral *** note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the legendre basis introduced by Jie Shen for Cartesian *** form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary ***,the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined,proving in a rigorous way one of the main advantages of the proposed radial bases.

关键词： Poisson equation cylindrical and spherical coordinates Fourier expansion spherical harmonics Jacobi polynomials spectral methods condition number

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Galerkin-laguerre Spectral Solution of Self-Similar Boundary layer Problems

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Communications in Computational Physics 2012年第10期12卷 1329-1358页

作者： F.Auteri l.quartapelle Politecnico di Milano Dipartimento di Ingegneria AerospazialeVia La Masa 3420156MilanoItaly

In this work the laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer *** original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown.A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer *** accuracy and efficiency of the laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld *** pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

关键词： laguerre polynomials semi-infinite interval boundary layer theory Falkner-Skan equation Cooke equation Orr-Sommerfeld equation linear stability of parallel flows

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Spectral Elliptic Solvers in a Finite Cylinder

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Communications in Computational Physics 2009年第2期5卷 426-441页

作者： F.Auteri l.quartapelle Politecnico di Milano Via La Masa 3420156 MilanoItaly

New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented.A purely variational(no collocation)formulation of the problem is adopted,based on Fourier series expansion of the angular dependence and legendre polynomials for the axial dependence.A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet *** Dirichlet boundary conditions are enforced by a discrete lifting and the vector problem is solved by means of a classical uncoupling *** the considered formulation,boundary conditions on the axis of the cylindrical domain are never mentioned,by *** solution algorithms for the scalar equations are based on double diagonalization along the radial and axial *** spectral accuracy of the proposed algorithms is verified by numerical tests.

关键词： Spectral elliptic solvers Dirichlet and Neumann conditions cylindrical coordinates legendre and Jacobi polynomials uncoupled vector problem

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Navier-Stokes Spectral Solver in a Finite Cylinder

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Communications in Computational Physics 2010年第8期8卷 663-689页

作者： F.Auteri M.Biava l.quartapelle Politecnico di Milano Dipartimento di Ingegneria AerospazialeVia La Masa 3420156 MilanoItaly

A primitive variable spectral method for simulating incompressible viscous flows inside a finite cylinder is *** element of originality of the proposed method is that the radial discretization of the Fourier coefficients depends on the Fourier mode,its dimension decreasing with the increase of the azimuthal modal *** principle was introduced independently by Matsushima and Marcus and by Verkley in polar coordinates and is adopted here for the first time to formulate a 3D cylindrical Galerkin projection method.A second element of originality is the use of a special basis of Jacobi polynomials introduced recently for the radial dependence in the solution of Dirichlet *** this basis the radial operators are represented by matrices of minimal sparsity-diagonal stiffness and tridiagonal mass-provided here in closed form for the first time,and lead to a Helmholtz operator characterized by a favorable condition ***,a new method is presented for eliminating the singular behaviour of the solution originated by the rotation of the lid with respect to the cylindrical *** to these elements,the resulting Navier-Stokes spectral solver guarantees the differentiability to any order of the solution in the entire computational domain and does not suffer from the time-step stability restriction occurring in spectral methods with a point clustering close to the *** test examples are offered that demonstrate the spectral accuracy of the solution method under different representative conditions.

关键词： Navier-Stokes equations finite cylindrical domain spectral methods Jacobi and legendre polynomials primitive variables projection method

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