A wavelet multi-resolution enabled interpolation Galerkin method for two-dimensional solids

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

doi:10.1016/j.enganabound.2020.04.007

Cited by 13 publications

(10 citation statements)

References 43 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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“…This allows the construction of a wavelet multiresolution approximation format that can suppress numerical instability at the boundaries and perform interpolation at any given point. [122,152] To construct a wavelet multiresolution interpolation that approximates a continuous function, f (x), defined on the region Ω, we first generate a set of completely uniform nodes x n = (k∕2 j 0 , l∕2 j 0 ), which includes all (k, l) satisfying Ω n ∩ Ω ≠ 0, where Ω n is a square with x n as the center and (𝛾 − 1)∕2 j 0 −1 as the side length, and 𝛾 is the order of the wavelet used, as shown in Figure 3. Additional nodes can be added at arbitrary positions.…”

Section: Interpolating Galerkin Methodsmentioning

confidence: 99%

“…, where the function 𝜌(n) ≥ j 0 returns the smallest integer that makes both k(n) = 2 𝜌(n) x n and l(n) = 2 𝜌(n) y n integers. [122,152] Based on this node distribution, a multiresolution approximation of f(x) can be expressed as: [133,170] f (x) ≈ P…”

Section: Interpolating Galerkin Methodsmentioning

confidence: 99%

“…Additionally, due to the majority of elements in its lower triangular part being zero, the computational effort required to compute its inverse matrix is negligible compared to solving the system of algebraic equations. After obtaining the inverse matrix of T, denoted as T −1 , the expansion coefficients, d n , can be represented by the node values, f(x n ), that is d = T −1 f , substituting it into Equation ( 47) yields [122,152]…”

Section: Interpolating Galerkin Methodsmentioning

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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“…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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A wavelet multi-resolution enabled interpolation Galerkin method for two-dimensional solids

Cited by 13 publications

References 43 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

Application of Wavelet Methods in Computational Physics

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

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