A numerical study of stationary solution of viscous Burgers’ equation using wavelet

Mittal, R. C.; Kumar, Sumit

doi:10.1080/00207160802322290

Cited by 7 publications

(4 citation statements)

References 18 publications

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“…[17,[70][71][72][73][74][75][76][77] These Coiflet-based WGMs have been widely acknowledged by peers, and have been directly cited and applied in the analysis of problems such as the vibration control of large-scale space structures, [78,79] the time-varying inhomogeneous electromagnetic problems, [80] non-homogeneous electric large-raywaveguide problems, [81] stochastic thermodynamics problems, [82] as well as material mechanics, [83] and fluid mechanics. [84][85][86][87][88] Although the Generalized WGM has achieved certain success in various fields, particularly in linear problems where it has demonstrated higher accuracy than the conventional Galerkin method, it encounters a bottleneck in solving nonlinear problems. In nonlinear problems, the standard WGM requires computations of the multiple integral of the product of scaling functions and their multiple derivatives, known as multiple connection coefficients.…”

Section: Wavelet Galerkin Methodsmentioning

confidence: 99%

“…[ 17,70–77 ] These Coiflet‐based WGMs have been widely acknowledged by peers, and have been directly cited and applied in the analysis of problems such as the vibration control of large‐scale space structures, [ 78,79 ] the time‐varying inhomogeneous electromagnetic problems, [ 80 ] non‐homogeneous electric large‐raywaveguide problems, [ 81 ] stochastic thermodynamics problems, [ 82 ] as well as material mechanics, [ 83 ] and fluid mechanics. [ 84–88 ]…”

Section: Introduction To Wavelet Numerical 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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Section: Wavelet Galerkin Methodsmentioning

confidence: 99%

Section: Introduction To Wavelet Numerical Methodsmentioning

confidence: 99%

Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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“…Wavelet Galerkin method has been in [161] for the numerical stationary solution of viscous Burgers equation (1) on interval (0, 1) with homogeneous neumann boundary conditions…”

Section: Survey Of Different Techniquesmentioning

confidence: 99%

Contemporary review of techniques for the solution of nonlinear Burgers equation

Dhawan

Kapoor

Kumar

et al. 2012

Journal of Computational Science

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“…It is important to note that wavelet bases have successfully been used in solving elliptic and parabolic PDEs and proved to be beneficial in developing adaptive schemes with optimal complexity and convergence rates, see, for example, . In addition, wavelet bases have been used in the context of finite difference and boundary element methods to solve physical problems (see for a survey on the application of wavelets in flow simulations).…”

Section: Introductionmentioning

confidence: 99%

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

Minbashian

Adibi

Dehghan

2017

Numerical Methods Partial

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This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017

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A numerical study of stationary solution of viscous Burgers’ equation using wavelet

Cited by 7 publications

References 18 publications

Application of Wavelet Methods in Computational Physics

Application of Wavelet Methods in Computational Physics

Contemporary review of techniques for the solution of nonlinear Burgers equation

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

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