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

Minbashian, Hadi; Adibi, Hojatollah; Dehghan, Mehdi

doi:10.1002/num.22180

Cited by 9 publications

(4 citation statements)

References 40 publications

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“…Their work opens perspectives for studies of nonlinear hyperbolic conservation laws using adaptive Galerkin discretizations. In addition, Minbashian et al [43] developed an adaptive wavelet SUPG method for hyperbolic conservation laws and performed a post processing using a denoising technique based on a minimization formulation to reduce Gibbs oscillations. Compared with the wavelet Galerkin methods, the collocation ones are more efficient.…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 99%

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

Yang¹,

Wang²,

Liu³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…For practical transient simulations with complex geometries and multiple timescales, adaptive procedure in both space and time are widely applied to find the approximate solution of the corresponding model (Abbaszadeh and Dehghan, 2020; Dehghan, 2001; Dehghan and Mohammadi, 2015, 2020; Ebrahimijahan et al , 2021; Kamranian et al , 2017; Minbashian et al , 2017). In this paper, in particular, we only focus on the time adaptive and aim to develop the adaptive time-stepping procedure for general purpose of transient simulations.…”

Section: Introductionmentioning

confidence: 99%

On the application of the GS4-1 framework for fluid dynamics and adaptive time-stepping via a universal A-posteriori error estimator

Wang

Xie

Yin

et al. 2022

HFF

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Purpose The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations. Design/methodology/approach The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder. Findings The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold. Originality/value To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

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“…Approximation of functions in L 2 (R) in wavelet basis with elements having compact support invented roughly thirty years 25 ago has a significant impact in many areas of pure and applied mathematics, [26][27][28][29][30][31][32][33][34][35][36] physics, 37 and engineering. 38,39 This is due to the elegant role of mathematical microscopy in the analysis of smoothness of function.…”

Section: Introductionmentioning

confidence: 99%

An efficient interpolating wavelet collocation scheme for quasi‐exactly solvable Sturm–Liouville problems in ℝ+

Singh

Saha

Banik

et al. 2022

Math Methods in App Sciences

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This investigation is an attempt to obtain a highly accurate approximation of the spectrum of Sturm–Liouville problems in ℝ+ by representing the unknown solution of the model in the interpolating wavelet basis of L2false(ℝfalse). To accomplish the goal, the domain ℝ+ has been stretched to ℝ to avoid the additional care of the elements in the basis containing boundary point 0. In addition, such transformation may judiciously be utilized to eliminate (up to quadratic) the singularity of the equation. The equation in the new variable has been subsequently transformed into a generalized matrix eigenvalue problem by approximating the new (unknown) function in an appropriate (truncated) basis comprising interpolating scale functions generated by scale functions in Daubechies family. The (interpolating wavelet‐collocation) scheme developed here has been applied to some solvable and quasi‐exactly solvable Sturm–Liouville problems in ℝ+ appearing in quantum mechanical modeling in flat and curved spaces. It is observed that the approximation of eigenfunctions in the (compact support) interpolating wavelet basis obtained by using the collocation method can be reliably used to reveal a hidden spectrum of quasi‐exactly solvable Sturm–Liouville problems in ℝ+ with high accuracy.

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

Cited by 9 publications

References 40 publications

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

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

On the application of the GS4-1 framework for fluid dynamics and adaptive time-stepping via a universal A-posteriori error estimator

An efficient interpolating wavelet collocation scheme for quasi‐exactly solvable Sturm–Liouville problems in ℝ+

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