Analysis and Application of Fourier–Gegenbauer Method to Stiff Differential Equations

Vozovoi, L.; Israeli, M.; Averbuch, Amir

doi:10.1137/s0036142994263591

Cited by 26 publications

(22 citation statements)

References 6 publications

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“…Also, it seems that global-local and local-local are producing similar errors for the same choice of A and m. Considering that the effective N (the number of collocation points) is smaller for the local-local case, hence round off errors should also be smaller, it seems that local-local should be used if possible (see also the discussions in [9] In Fig. 4 we show similar errors for the Procedure B.…”

Section: Numerical Resultsmentioning

confidence: 54%

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

and

1995

Numerische Mathematik

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The paper presents a method to recover exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of an approximation to the interpolation polynomial (or trigonometrical polynomial). We show that if we are given the collocation point wdues (or an highly accurate approximation) at the Gauss or Gauss-Lobatto points, we can reconstruct an uniform exponentially convergent approximation to the function f(x) in any sub-interval of analyticity. The proof covers the cases of Fourier, Chebyshev, Legendre, and more general Gegenbauer collocation methods.

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Section: Numerical Resultsmentioning

confidence: 54%

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

and

1995

Numerische Mathematik

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“…Em [13], uma suavização preliminar (denominada subtração polinomial por [5], p. 77) foi empregada para melhorar a convergência das aproximações de FG de funções altamente oscilatórias. Um procedimento para lidar com soluções envolvendo fortes gradientes nos contornos do domínioé apresentado em [12]. Observamos também que técnicas de subtração de singularidades (ver, por exemplo, [1]) também podem ser implementadas para reduzir erros de regularização e acelerar a convergência das expansões de Gegenbauer.…”

Section: Resolução Da Eq De Poisson Por Fgunclassified

“…Vozovoi e co-autores aplicaram este método para resolver problemas a valores no contorno com funções forçantes suaves e não-periódicas ( [13]) e equações diferenciais 'stiff' ( [12]). No presente trabalho, propomos um método de Fourier-Gegenbauer (FG) adequado para a resolução numérica das equações de Poisson/Helmholtz unidimensionais.…”

Section: Introductionunclassified

Um Método Unidimensional de Fourier-Gegenbauer para a Resolução da Equação de Helmholtz

Oliveira¹,

Eying²

2004

TEMA - Tendências Em Matemática Aplicada E Computacional

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“…Several numerical techniques can be used for the evaluation of derivatives with high accuracy. One is based on the Fourier-Gegenbauer method [12]. Another, which is much simpler but which is a sufficiently accurate alternative, is described in Appendix A.…”

Section: Nonperiodic Boundary Conditionsmentioning

confidence: 99%

“…1(5)]). A particular version of this technique in one dimension was implemented in [12] for the solution of stiff differential equations with high accuracy.…”

Section: Nonperiodic Boundary Conditionsmentioning

confidence: 99%

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

Averbuch¹,

Israeli²,

Vozovoi³

1998

SIAM J. Sci. Comput.

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Abstract. In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems.The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms.The solution of a boundary value problem is obtained in a series form in O(N log N ) floating point operations, where N 2 is the number of grid nodes. Evaluating the solution at all N 2 interior points requires O(N 2 log N ) operations.Key words. boundary value problem, Poisson equation, rectangular region, spectral method, corner discontinuities, polynomial subtraction AMS subject classifications. 35J05, 45L10, 65P05PII. S1064827595288589 1.Introduction. An important step in the development of fast numerical solvers for elliptic equations in complicated domains is an algorithm for the solution of boundary value problems for constant coefficient elliptic equations in rectangular regions. After solving a nonhomogeneous equation with some "convenient" boundary conditions, one must solve, in a correction step, the homogeneous problem with specified (Dirichlet or Neumann) boundary conditions. As a resulting boundary distribution may not satisfy the required compatibility conditions at the corner points, singularities may arise in the process of the solution, thus destroying the convergence rate even if the final solution is smooth.When the computational region is discretized on a N × N grid using some loworder (finite difference or finite element) scheme, the resulting system of linear algebraic equations is represented by a sparse matrix. Such a matrix can be inverted in O(N 2 ) (or O(N 2 log 2 N ) arithmetic operations using a "fast solver" [4]. However, since the method is of low order, the resolution N must be very large if a high accuracy is desired.Application of high-order (pseudo) spectral methods, based on global expansions into orthogonal polynomials, e.g., Chebyshev polynomials, to the solution of elliptic equations, results in a full matrix problem. The cost of inverting such a matrix using the best current algorithms is O(N 3 ) operations [2]. Besides, the accuracy decreases considerably as the dimension N 2 of the system grows due to accumulation of roundoff errors.When using the Chebyshev method, the fast computation of the expansion coefficients requires that the problem be discretized on a nonuniform grid. For timedependent problems, when elliptic equations arise due to a time-discretization pro-

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Analysis and Application of Fourier–Gegenbauer Method to Stiff Differential Equations

Cited by 26 publications

References 6 publications

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function

Um Método Unidimensional de Fourier-Gegenbauer para a Resolução da Equação de Helmholtz

A Fast Poisson Solver of Arbitrary Order Accuracy in Rectangular Regions

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