Spectral and finite difference solutions of the Burgers equation

Basdevant, C.; Deville, Michel; Haldenwang, Pierre; Lacroix, Jean; Quazzani, J.

doi:10.1016/0045-7930(86)90036-8

Cited by 203 publications

(144 citation statements)

References 7 publications

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“…To compute the additional temporal level we may use, for instance, a single-step method to integrate equations (5.6). An experimental analysis of several IMEX schemes, including the one used in (5.7), was carried out in [4] and [5]. Recently, the author also used this IMEX scheme to approximate the solutions of a weakly-nonlinear, weakly-dispersive Boussinesqtype system with highly-variable coefficients [11].…”

Section: Numerical Resultsmentioning

confidence: 99%

Existence and Numerical Approximation of Solutions of an Improved Internal Wave Model

Grajales

2014

Mathematical Modelling and Analysis

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Abstract. In this paper we establish local existence of solutions for a new model to describe the propagation of an internal wave propagating at the interface of two immiscible fluids with constant densities, contained at rest in a long channel with a horizontal rigid top and bottom. We also introduce a spectral-type numerical scheme to approximate the solutions of the corresponding Cauchy problem and perform a complete error analysis of the semidiscrete scheme.

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Section: Numerical Resultsmentioning

confidence: 99%

Existence and Numerical Approximation of Solutions of an Improved Internal Wave Model

Grajales

2014

Mathematical Modelling and Analysis

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“…The analytic result quoted in [2] is m = 1.5200516 occuring at art = 1.6037. We have found analogously to what was found in [9] that the coordinate system which yields the most accurate calculation of m is not perfectly predicted by the adaptive procedure.…”

Section: Mappingmentioning

confidence: 99%

“…We consider the Burgers equation The viscosity coefficient v = .01/7r. This problem has been used as test case for a variety of spectral, pseudo-spectral and finite difference methods [2,9]. As t increases the solution develops a very steep gradient at the x = 0.…”

Section: Mappingmentioning

confidence: 99%

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Mappings and accuracy for Chebyshev pseudo-spectral approximations

Bayliss

Turkel

1992

Journal of Computational Physics

107

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The effect of mappings on the approximation, by Chebyshev collocation, of functions which exhibit localized regions of rapid variation is studied. A general strategy is introduced whereby mappings are adaptively constructed which map specified classes of rapidly varying functions into low order polynomials which can be accurately approximated by Chebyshev polynomial expansions. A particular family of mappings constructed in this way is tested on a variety of rapidly varying functions similar to those occurring in approximations. It is shown that the mapped function can be approximated much more accurately by Chebyshev polynomial approximations than in physical space or where mappings constructed from other strategies are employed. Introduction.One of the major difficulties in the application of Chebyshev pseudo-spectral methods, or other spectral methods, to the solution of partial differential equations, is in the approximation of functions which exhibit localized regions of rapid variation.

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“…A comparison of numerical solutions of the Burgers equation by the spectral method and finite difference method (FDM) can be found in [3]. Since the birth in the 1980s of wavelet methods, several works ( [18,14]) have been devoted to the numerical solution of the Burgers equation using these methods.…”

Section: Introductionmentioning

confidence: 99%

Wavelet-Taylor Galerkin Method for the Burgers Equation

Kumar

Mehra

2005

Bit Numer Math

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Abstract.In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L∞ errors relative to the analytical solution together with the numerical solution are reported. (2000): 65M70. AMS subject classification

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Spectral and finite difference solutions of the Burgers equation

Cited by 203 publications

References 7 publications

Existence and Numerical Approximation of Solutions of an Improved Internal Wave Model

Existence and Numerical Approximation of Solutions of an Improved Internal Wave Model

Mappings and accuracy for Chebyshev pseudo-spectral approximations

Wavelet-Taylor Galerkin Method for the Burgers Equation

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