A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment

Liu, Xiaojing; Liu, G. R.; Wang, Jizeng; Zhou, Youhe

doi:10.1007/s00466-019-01691-6

Cited by 24 publications

(33 citation statements)

References 52 publications

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“…However, the proposed method was only verified by nonlinear wave problems in regular spatial domains. For a future study, one may combine the algorithm for wavelet approximation of functions bounded in irregular domain, as we have developed previously [42,43], into the proposed method to solve real engineering problems with irregular shapes/domains.…”

Section: Discussionmentioning

confidence: 99%

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

et al. 2021

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A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

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Section: Discussionmentioning

confidence: 99%

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

et al. 2021

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“…For general practical problems, Ω is a finite domain. We apply the boundary extension technique in our previous study based on the Lagrange interpolation to remove local errors induced by a loss of information outside the domain [29,30]. For all functions in 2 () L Ω , the modified wavelet approximation at the resolution level J can be written as…”

Section: Approximation Of Functions On a Finite Domainmentioning

confidence: 99%

Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws

Yang¹,

Wang²,

Liu³

et al. 2023

Preprint

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A system of wavelet collocation upwind schemes is constructed for solving hyperbolic conserva-tion laws based on a class of interpolation wavelets. The bias magnitude and symmetry factor are defined to depict the asymmetry of the adopted scaling basis function in wavelet theory. Effects of characteristics of the scaling functions on the schemes are explored based on numerical tests and Fourier analysis. The numerical results reveal that the stability of the constructed scheme is af-fected by the smoothness order, N, and the asymmetry of the scaling function. The dissipation analysis suggests that schemes with Neven have negative dissipation coefficients, leading to unstable behaviors. Only scaling functions with Nodd and bias magnitude of 1 can be used to construct stable upwind schemes due to the non-negative dissipation coefficients. Resolution of the wavelet scheme tends to the spectral resolution as the order of accuracy of the scheme in-creases.

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“…Wang et al 23 have constructed finite element multiwavelets using linear Lagrange and cubic Hermite scaling functions and the lifting scheme in order to achieve scale decoupling in the static analyses of rod and beam elements. Liu et al 24 have developed explicitly a stable wavelet interpolant which they have applied to a proposed wavelet multiresolution interpolation Galerkin method for targeted local enrichment in solving 2D static problems with complex load cases and boundaries.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids

Nastos

Saravanos

2021

Numerical Meth Engineering

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The multiresolution capability provided by the family of Daubechies wavelets is exploited to develop a new computational approach, termed as multiresolution finite wavelet domain method for the fast and hierarchical prediction of transient dynamic problems with focus on wave propagation in one-and two-dimensional structural configurations. In the developed method, both scaling and wavelet functions are employed as basis functions for the approximation of displacements in the governing dynamic equations. Because of the orthogonal and the multiresolution properties of Daubechies wavelets, a two-scale system of mass uncoupled dynamic equations, representing the coarse and the fine spatial wave component solutions is derived. Numerical results concerning wave propagation in one-and two-dimensional elastic structures demonstrate substantial computational gains of the method in comparison with other well-established numerical methods. Moreover, additional benefits of the involved coarse and fine solutions enabling the analysis and characterization of the resultant wave response are revealed and discussed.

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A wavelet multiresolution interpolation Galerkin method for targeted local solution enrichment

Cited by 24 publications

References 52 publications

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

Stability and Resolution Analysis of the Wavelet Collocation Upwind Schemes for Hyperbolic Conservation Laws

Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids

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