The Spectrum of the Chebyshev Collocation Operator for the Heat Equation

Gottlieb, David; Lustman, Liviu

doi:10.1137/0720063

Cited by 99 publications

(65 citation statements)

References 3 publications

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“…In Section 3 we use the results obtained for the hyperbolic equations to show that for the heat equation the second-derivative matrices, corresponding to the Neumann conditions with the new approach, have real and negative eigenvalues. The analogous result for the classical way to impose boundary conditions was previously proven in [8].…”

Section: Introductionmentioning

confidence: 51%

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A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations

Funaro¹,

Gottlieb²

1988

Mathematics of Computation

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Abstract.A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.Introduction.The common practice in applying pseudospectral methods to partial differential equations is to satisfy the equation at the interior nodes and to impose the boundary condition at the boundary. This procedure does not take into consideration that the differential equation is satisfied at points arbitrarily close to the boundary. In [4], one of the authors discussed the advantages of imposing a combination of boundary conditions and the equation itself at the boundary nodes, for Chebyshev approximations of the Laplace equation with Neumann conditions. Here we analyze the same idea applied to the linear hyperbolic equation

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Section: Introductionmentioning

confidence: 51%

“…In order to show that the roots of g(p) and h(p) are real negative and distinct, we use the notion of a positive pair (see [5] and [8]). Two polynomials form a positive pair if their roots are real negative and interlaced.…”

Section: Boundary Conditions For Elliptic Equationsmentioning

confidence: 99%

A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations

Funaro¹,

Gottlieb²

1988

Mathematics of Computation

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“…As proved in [9], they are real, distinct and negative. We now display l/vi, where ui are the eigenvalues of (6.14) (6.15) for TV = 8:…”

Section: The Y-dependentmentioning

confidence: 99%

“…Using the results in [9] and looking at the eigenvalue problem There are (N -l)(M -1) eigenvalues and the same number of linearly independent eigenfunctions of (7.5)-(7.7), and it is easy to see that the eigenfunctions are products of the one-dimensional eigenfunctions, i.e., *%'M,(x,y) = *k(y)*i(x), where $k(y) is an eigenfunction of (6.5)-(6.6) and $;(x) is an eigenfunction of (7.11)-(7.12).…”

Section: The Y-dependentmentioning

confidence: 99%

The Spectrum and the Stability of the Chebyshev Collocation Operator for Transonic Flow

Fishelov¹

1988

Mathematics of Computation

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Abstract.The extension of spectral methods to the small disturbance equation of transonic flow is considered.It is shown that the real parts of the eigenvalues of its spatial operator are nonpositive. Two schemes are considered; the first is spectral in the x and y variables, while the second is spectral in x and of second order in y. Stability for the second scheme is proved. Similar results hold for the two-dimensional heat equation.

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“…Since Fourier analysis is used in y, we consider the region -oo < y < oo. where u/ç are the eigenvalues of (6.5)-(6.6), which are real, distinct and negative by [9]. Since / **(yi) \ \*k(yM-i)J fc=i,...,M-i are linearly independent, we have…”

Section: The Y-dependentmentioning

confidence: 99%

The spectrum and the stability of the Chebyshev collocation operator for transonic flow

Fishelov¹

1988

Math. Comp.

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The Spectrum of the Chebyshev Collocation Operator for the Heat Equation

Cited by 99 publications

References 3 publications

A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations

A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations

The Spectrum and the Stability of the Chebyshev Collocation Operator for Transonic Flow

The spectrum and the stability of the Chebyshev collocation operator for transonic flow

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