Stability and Convergence analysis of a Crank-Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

2023/11/11 by Dwivedi, Mukul, Sarkar, Tanmay · 1 citation
#65M60 #FOS: Mathematics #Numerical Analysis (math.NA) #Primary: 35Q53 #Secondary: 65M12

paper · doi:10.48550/arxiv.2311.06589

Abstract

In this paper we study the convergence of a fully discrete Crank-Nicolson Galerkin scheme for the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and non-linear convection terms. Our proof relies on the Kato type local smoothing effect to estimate the localized H^α/2-norm of the approximated solution, where α∈ [1,2). We demonstrate that the scheme converges strongly in L²(0,T;L²_loc(ℝ)) to a weak solution of the fractional KdV equation provided the initial data in L²(ℝ). Assuming the initial data is sufficiently regular, we obtain the rate of convergence for the numerical scheme. Finally, the theoretical convergence rates are justified numerically through various numerical illustrations.

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• A numerical method for the fractional Zakharov-Kuznetsov equation

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