A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

Bhatt, Harish; Chowdhury, Abhinandan

doi:10.1080/00207160.2020.1814262

Cited by 14 publications

(5 citation statements)

References 41 publications

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“…The initial and boundary conditions are taken from this exact solution. The computational domain is considered as ½x 0 , x N ¼ À50, 50 ½ and with different k: Let us take the values of the parameters e ¼ 5, r ¼ 0:5 ffiffiffiffiffiffiffiffi 11 19 À Á q , c ¼ À25 to compare the approximate solutions obtained by the present method with the results computed by ( Bhatt & Chowdhury, 2019;Lai & Ma, 2009;Mittal & Arora, 2010;Singh et al, 2018;Zarebnia & Parvaz, 2013 ). Comparison of GRE and L 1 with others at different values of t and N are reported in Tables 2 and 4.…”

Section: Numerical Examples and Discussionmentioning

confidence: 99%

“…Besides, a large numbers of other numerical schemes have been devoted to solving KSE such as backward finite difference scheme proposed by Akrivis and Smyrlis (2004), robust Stackelberg controllability used by Montoya and Breton (2020), new iterative technique applied by Abed (2020), the semi-analytical scheme implemented by Shah, Khan, Baleanu, Kumam, and Arif (2020), discontinuous Galerkin scheme applied by Xu and Shu (2006), tanh-function scheme proposed by Fan (2000), Chebyshev spectral collocation method implemented by Khater and Temsah (2008), compact finite difference method employed by Singh, Arora, and Kumar (2018), compact fourth-order implicit-explicit Runge-Kutta method developed by Bhatt and Chowdhury (2019), compact finite difference method with orthogonal decomposition adopted by Zhang, Zhang, and Ding (2019) and lattice Boltzmann scheme implemented by Lai and Ma (2009). Each method used to solve KSE, still involves certain drawbacks like high arithmetic computations, lower accuracy in terms error, difficulty for computer programming and limitations of certain special cases.…”

Section: Introductionmentioning

confidence: 99%

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Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Jena

Gebremedhin

2021

Arab Journal of Basic and Applied Sciences

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In this article, a nonic B-spline collocation approach is applied to solve the approximate solution of Kuramoto-Sivashinsky equation (KSE). Here the nonlinear term of KSE is linearizing using Taylor series technique. The main objective of this study is to provide a new class of approach by applying some possible higher-order derivatives rather than lower order derivatives of the nonic B-spline on the boundary conditions of KSE in finding the additional constraints, which help us to obtain a unique solution of the problem. The stability analysis is investigated using the Von-Neumann scheme and the proposed scheme is found to be unconditionally stable. The efficiency and accuracy of the proposed approach are demonstrated by two numerical tests and the current results are compared with the exact solution and result of others in the literature.

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Section: Numerical Examples and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Jena

Gebremedhin

2021

Arab Journal of Basic and Applied Sciences

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“…In the previous years, researchers have extensively studied the KS equation using a diverse array of numerical techniques. Ersoy [41] solved KS equation via Galerkin method, Denis S G. [42] used numerical simulations to study how introducing advection affects localized patterns in a system governed by a modified KS equation with frozen disorder, Shah et al [43] presented A semi-analytical method to solve family of KSE, Bhatt and Chowdhury [44] developed a numerical scheme combining a fourth-order Runge-Kutta-based implicitexplicit scheme in time and a compact higher-order finite difference scheme in space, Jena and Gebremedhin [45] applied a nonic B-spline collocation approach to solve the KS equation and they used the Taylor series technique to linearize the nonlinear term of the KS equation during the solution process, Mkhize et al [46] introduced heptic Hermite basis functions, employed them in the orthogonal collocation on finite elements (OCFE) method, and used this approach to solve the GKSE.…”

Section: Introductionmentioning

confidence: 99%

An efficient scheme for solving nonlinear generalized kuramoto-sivashinksy processes

Mahmood,

Tahir,

Jwamer

2023

Phys. Scr.

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In this work, we investigate the numerical solution of generalized Kuramoto-Sivashinksy (GKS) problems based on the collocation of the quantic B-spline (QBS) and high-order strong stability-preserving Runge–Kutta (SSPRK54) scheme. When considering nonlinear parts that lose real features, we address the issue without resorting to any transformations or linearization. The efficiency and accuracy of our proposed technique are evaluated using a variety of illustrative examples. The numerical results show that our approach captured the natural behaviour of the problems well and consumed less storage space.

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“…The splitting technique was used, which resulted in a coupled system of equations. Bhatt and Chowdhury 26 introduced a CFDS of fourth‐order in space and implicit–explicit Runge–Kutta scheme of fourth‐order in time domain to solve this equation. Larios and Yamazaki 27 proposed a reduced 2D model which is globally well posed.…”

Section: Introductionmentioning

confidence: 99%

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

Shallu

Kukreja

2021

Math Methods in App Sciences

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In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation.

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A high-order implicit–explicit Runge–Kutta type scheme for the numerical solution of the Kuramoto–Sivashinsky equation

Cited by 14 publications

References 41 publications

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

Numerical treatment of Kuramoto-Sivashinsky equation on B-spline collocation

An efficient scheme for solving nonlinear generalized kuramoto-sivashinksy processes

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

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