A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation

Oruç, Ömer; Bulut, Fatih; Esen, Alaattin

doi:10.1007/s10910-015-0507-5

Cited by 71 publications

(36 citation statements)

References 31 publications

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“…After this achievement researchers have been using Haar wavelets to obtain numerical solutions of differential equations because of their simplicity and computational features. Recently, many authors have used Haar wavelet method for solving ordinary and partial differential equations [20][21][22][23][24][25][26][27][28][29][30][31]. Especially high order pdes like KdV and fractional coupled KdV equations are considered in [32,33].…”

Section: Haar Waveletsmentioning

confidence: 99%

A numerical treatment based on Haar wavelets for coupled KdV equation

Oruç¹,

Bulut

Esen

2017

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In this paper, numerical solutions of one dimensional coupled KdV equation has been investigated by Haar Wavelet method. Time derivatives given in this equation are discretized by finite differences and nonlinear terms appearing in the equations are linearized by some linearization techniques and space derivatives are discretized by Haar wavelets. For examining performance of the proposed method, single soliton solution and conserved quantities of some test problems are used. Also error analysis of numerical scheme is investigated and numerical results are compared with some results already existing in the literature.

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Section: Haar Waveletsmentioning

confidence: 99%

A numerical treatment based on Haar wavelets for coupled KdV equation

Oruç¹,

Bulut

Esen

2017

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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“…Unfortunately, the number of relevant degrees of freedom is finite in truncated field computations. 7,8 While improved hardware and data-oriented strategies such as parallel computing have made great strides in the past few decades, memory costs and computational expenses associated with massively parallel uniform grid simulations represent a great motivation for utilizing adaptive grid strategies for complex physics exhibiting sharp gradients. Multiresolution techniques bridge the gap between both spectral and finite difference/volume approaches by providing a framework for the identification of high-frequency regions in the solution field through its hybrid behavior, ie, being localized in both space and scale.…”

Section: Discussionmentioning

confidence: 99%

“…7,8 While improved hardware and data-oriented strategies such as parallel computing have made great strides in the past few decades, memory costs and computational expenses associated with massively parallel uniform grid simulations represent a great motivation for utilizing adaptive grid strategies for complex physics exhibiting sharp gradients. The high degree of localization makes the use of spectral methods inefficient since global expansion fails to represent the solution accurately at lower degrees of resolution.…”

mentioning

confidence: 99%

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

Maulik

San

Behera

2018

Numerical Methods in Fluids

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Summary We put forth a dynamic computing framework for scale‐selective adaptation of weighted essential nonoscillatory (WENO) schemes for the simulation of hyperbolic conservation laws exhibiting strong discontinuities. A multilevel wavelet‐based multiresolution procedure, embedded in a conservative finite volume formulation, is used for a twofold purpose. (i) a dynamic grid adaptation of the solution field for redistributing grid points optimally (in some sense) according to the underlying flow structures, and (ii) a dynamic minimization of the in built artificial dissipation of WENO schemes. Taking advantage of the structure detection properties of this multiresolution algorithm, the nonlinear weights of the conventional WENO implementation are selectively modified to ensure lower dissipation in smoother areas. This modification is implemented through a linear transition from the fifth‐order upwind stencil at the coarsest regions of the adaptive grid to a fully nonlinear fifth‐order WENO scheme at areas of high irregularity. Therefore, our computing algorithm consists of a dynamic grid adaptation strategy, a scale‐selective state reconstruction, a conservative flux calculation, and a total variation diminishing Runge‐Kutta scheme for time advancement. Results are presented for canonical examples drawn from the inviscid Burgers, shallow water, Euler, and magnetohydrodynamic equations. Our findings represent a novel direction for providing a scale‐selective dissipation process without a compromise on shock capturing behavior for conservation laws, which would be a strong contender for dynamic implicit large eddy simulation approaches.

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“…Haq et al in [11] formulated simple classical radial basis functions (RBFs) collocation method for the numerical solution of the non-linear dispersive and dissipative Burgers equation. Both Orac et al in [12] and Lepik in [13] investigated the numerical solutions using Haar wavelet. Inan and Bahadir described implicit exponential difference method in two cases: finite and fully finite [14].…”

Section: Introductionmentioning

confidence: 99%

“…We take c 0 = 0.5 and β = 0.01. Table 3 shows the L ∞ errors by using CSC method and given methods in [9,12,15,16] for t = 2, 4 and 6 and N = 9 × 9. In Figures 8 and 9, we show the approximate solution and absolute error, respectively.…”

mentioning

confidence: 99%

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

Skandari

Mohammadizadeh

et al. 2019

Symmetry

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This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

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A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation

Cited by 71 publications

References 31 publications

A numerical treatment based on Haar wavelets for coupled KdV equation

A numerical treatment based on Haar wavelets for coupled KdV equation

An adaptive multilevel wavelet framework for scale‐selective WENO reconstruction schemes

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

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