Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

Liu, Xiaojing; Zhou, Youhe; Wang, Jizeng

doi:10.1007/s10483-022-2859-5

Cited by 7 publications

(5 citation statements)

References 42 publications

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“…[140,141] Chen and colleagues established WFEMs using interval B-spline wavelets and effectively applied them to the dynamic and static analysis of beam, plate, and shell structures. [3,117,[118][119][120][121][122][123][124][125][126][127][128][129][130][131] Their research demonstrated that the WFEM has high precision and can yield usable results with fewer nodes.…”

Section: Wavelet Finite Element Methodsmentioning

confidence: 99%

“…Currently, this approach has been successfully applied to stress analysis and fracture simulation of mechanical components with complex geometries, as well as the solution of nonlinear boundary layer problems, demonstrating superior comprehensive performance compared to traditional methods such as FEMs. [122][123][124]152,153] This wavelet multiresolution Galerkin method provides a viable solution to the key foundational challenges faced by waveletbased numerical methods in dealing with irregular domains, and it possesses several attractive advantages: 1). It does not require a mesh, including a background grid, making it a true meshless method; 2).…”

Section: Interpolating Galerkin Methodsmentioning

confidence: 99%

“…Due to the lack of analytical expressions and high oscillation of derivatives for compactly supported othogonal wavelets, conventional numerical integration techniques cannot easily obtain the connection coefficients. Currently, several approaches have been developed to address the handling of irregular domains in wavelet methods, including Wavelet FEM (WFEM), [115][116][117][118][119] VDM, [114,120,121] Interpolation WGM (IWGM), [122][123][124] and Wavelet Integral Collocation Method (WICM). [125][126][127][128]…”

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

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Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

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Section: Wavelet Finite Element Methodsmentioning

confidence: 99%

Section: Interpolating Galerkin Methodsmentioning

confidence: 99%

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

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Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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“…If is marked as the trouble one. Once all trouble nodes are recognized, a set of nodes are inserted at the high resolution level based on the algorithm proposed in our previous research [67] to improve the local approximation accuracy. The strategy with suitable parameters adds enough nodes in the adjacent zone of the trouble nodes to guarantee the accuracy for time evolution problems.…”

Section: Adaptive Multiresolution Wavelet Collocation Upwind Schemesmentioning

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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