A small time solutions for the KdV equation using Bubnov-Galerkin finite element method

Amein, N. K.; Ramadan, Mohamed A.

doi:10.1016/j.joems.2011.10.005

Cited by 13 publications

(6 citation statements)

References 8 publications

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“…The active use of solitons in the study and solution of nonlinear wave equations [12] describing physical phenomena in many areas [13] stimulated interest in methods for solving the KdV equation. The KdV equation was solved numerically by various methods, such as the Galerkin method [14][15][16], the collocation method [17,18], the finite element method [19][20][21], the finite-difference method [22][23][24][25][26][27][28][29][30], etc. The choice of one or another numerical solution method largely determines the quality of numerical modeling.…”

Section: Introductionmentioning

confidence: 99%

Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

Mazhukin¹,

Shapranov²,

Bykovskaya³

2020

MathMon

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A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with the parameter σ = 0.5 and the order of approximation O(Δt2 + Δx2) isshown. Completely implicit two-layer difference schemes with the parameter σ = 1 and O (Δt+ Δx2) are characterized by absolute stability with a low solution accuracy due to a highapproximation error. The family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter σ <0.5 prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.

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Section: Introductionmentioning

confidence: 99%

Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

Mazhukin¹,

Shapranov²,

Bykovskaya³

2020

MathMon

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“…The numerical stability of the temporal discretization is then established by the Courant-Friedrichs-Lewy (CFL) condition [6]. This type of approximation has been established for the KdV equation using multiple types of FE methods including discontinuous Galerkin (DG) method by multiple authors [7][8][9], the hybridized DG method of Samii et al [10], Galerkin methods using C 1 or higher bases [11][12][13][14][15], and spectral element methods [16,17]. Due to the success of the aforementioned methods and the ease of implementation of a CFL condition for stability of time stepping schemes, space-time FE methods have, to our best knowledge, not been applied to the KdV equation.…”

Section: Introductionmentioning

confidence: 99%

An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

Valseth,

Dawson

2020

Preprint

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We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov-Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H 1 , while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C 0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.

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“…A study based on cubic B-spline finite element method for the solution of the KdVE is suggested by Kapoor et al [18]. A Bubnov-Galerkin finite element method with quintic B-spline functions taken as element shape and weight functions is presented for the solution of the KdVE [19]. The paper deals with the numerical solution of the KdVE using quartic B-splines Galerkin method as both shape and weight functions over the finite intervals [20].…”

Section: Introductionmentioning

confidence: 99%

The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

Ersoy

Dağ

2015

Advances in Numerical Analysis

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A small time solutions for the KdV equation using Bubnov-Galerkin finite element method

Cited by 13 publications

References 8 publications

Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

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