Connection coefficients on an interval and wavelet solutions of Burgers equation

Lin, En-Bing; Zhou, Xu

doi:10.1016/s0377-0427(00)00562-8

Cited by 40 publications

(22 citation statements)

References 8 publications

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“…The resulting linear system is singular and we should add some (not necessarily nonhomogeneous) constraints to obtain the unique solution. Commonly, these constraints are constructed by using moment conditions (Romine and Peyton, 1997;Lin and Zhou, 2001). For arbitrary i, s, Eqn.…”

Section: H Akbarimentioning

confidence: 99%

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Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Akbari¹

2013

International Journal of Applied Mathematics and Computer Science

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We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2 −nN ) in the H 1 norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection coefficients. Due to the fast convergence rate, very good approximations are found at low levels and with low Coiflet degrees, hence the size of corresponding linear systems is small. Numerical experiments confirm these claims.

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Section: H Akbarimentioning

confidence: 99%

“…A problem that arises in the wavelet-Galerkin method is evaluation of these values on the interval (Lin and Zhou, 2001). The highly oscillatory nature of wavelet basis functions makes standard numerical quadrature of integrals near the boundary impractical.…”

mentioning

confidence: 99%

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Akbari¹

2013

International Journal of Applied Mathematics and Computer Science

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“…The integrals ⌫ pm j , ⍀ pm j , ␣ pm j are called the connection coefficients at level j (or the connection coefficients on the interval as defined in [7] for the orthonormal scaling functions). The following relations are obvious:…”

Section: The Setup Of the Algebraic Equationsmentioning

confidence: 99%

“…In order to calculate ⌫ pm 0 , ⍀ pm 0 , we recall the definition of the connection coefficients on the interval [0, 1] in [7]: Therefore, we only need to compute nonzero⌫ pm 0 ,⍀ pm 0 for Ϫ1 Յ p, m Յ 2. Since is a refinable function, i.e., Each of (3.10) is a system of linear homogeneous equations.…”

Section: Calculations Of the Connection Coefficients At Level J mentioning

confidence: 99%

Wavelets‐Galerkin scheme for a Stokes problem

Zhou

2003

Numerical Methods Partial

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“…The use of wavelets in Galerkin methods for the solution of partial differential equations often leads to computationally expensive schemes, although there are interesting possibilities to exploit the localization properties of the basis elements (see, e.g., [1][2][3]), also in combination with local refinement techniques [4,5]. High computational costs are mainly related to the connection coefficients in nonlinear equations, besides the solution costs.…”

Section: Introductionmentioning

confidence: 99%

Computational aspects of harmonic wavelet Galerkin methods and an application to a precipitation front propagation model

Barros

Peixoto

2011

Computers & Mathematics with Applications

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a b s t r a c tThis article is dedicated to harmonic wavelet Galerkin methods for the solution of partial differential equations. Several variants of the method are proposed and analyzed, using the Burgers equation as a test model. The computational complexity can be reduced when the localization properties of the wavelets and restricted interactions between different scales are exploited. The resulting variants of the method have computational complexities ranging from O(N 3 ) to O(N) (N being the space dimension) per time step. A pseudo-spectral wavelet scheme is also described and compared to the methods based on connection coefficients. The harmonic wavelet Galerkin scheme is applied to a nonlinear model for the propagation of precipitation fronts, with the front locations being exposed in the sizes of the localized wavelet coefficients.

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Connection coefficients on an interval and wavelet solutions of Burgers equation

Cited by 40 publications

References 8 publications

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Wavelets‐Galerkin scheme for a Stokes problem

Computational aspects of harmonic wavelet Galerkin methods and an application to a precipitation front propagation model

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