Convergence of a conservative Crank-Nicolson finite difference scheme for the KdV equation with smooth and non-smooth initial data

2023/12/22 by Mukul Dwivedi, Dwivedi, Mukul, Tanmay Sarkar +1
Mathematics · Physics and Astronomy · #65M06 #Advanced Mathematical Physics Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2312.14454

openalex publication_date 2023/12/22 · openalex created_date 2023/12/26 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the stability and convergence of a fully discrete finite difference scheme for the initial value problem associated with the Korteweg-De Vries (KdV) equation. We employ the Crank-Nicolson method for temporal discretization and establish that the scheme is L²-conservative. The convergence analysis reveals that utilizing inherent Kato's local smoothing effect, the proposed scheme converges to a classical solution for sufficiently regular initial data u₀ ∈ H³(ℝ) and to a weak solution in L²(0,T;L²_loc(ℝ)) for non-smooth initial data u₀ ∈ L²(ℝ). Optimal convergence rates in both time and space for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.

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