Abstract
In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods.
Highlights
We address this paper to the implementation of the Generalized Finite Difference Method (GFDM)to solve the complex Ginzburg–Landau equation, ∂U− (ν + iα)∆U + ( β + iμ)|U |2 U − γU = 0, x ∈ Ω, t > 0, ∂tU ( x, y, 0) = U0 ( x, y), x ∈ Ω,U ( x, y, t) = b( x, y, t), x ∈ ∂Ω, t > 0, (1)for some enough regular functions U0 ( x, y), b( x, y, t) in Ω × [0, ∞), where Ω ⊂ R2 is a bounded domain
In this paper we propose the Generalized Finite Difference Method to solve numerically (1) and determine the behavior of the discrete solution of the numerical scheme generated by that method
We present a comparison between the results obtained by using the GFDM in this paper and the ones obtained in [1,22]
Summary
In this paper we propose the Generalized Finite Difference Method to solve numerically (1) and determine the behavior of the discrete solution of the numerical scheme generated by that method This meshless method has been widely used since Lizska and Orkisz [9] and the explicit formulas of the method were derived by Benito, Gavete and Ureña [10,11,12]. The explicit formulae of the method allow us to obtain the discretization of the spatial partial derivatives Another advantage of the method, stated in [10] is the small number of nodes at each star (8 + 1) for 2D and (26 + 1) in 3D, which results in almost empty matrices.
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