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arXiv · 2312.14454

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

Mukul Dwivedi·Tanmay Sarkar
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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 $L^2$-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 $u_0 \in H^{3}(\mathbb{R})$ and to a weak solution in $L^2(0,T;L^2_{\text{loc}}(\mathbb{R}))$ for non-smooth initial data $u_0 \in L^2(\mathbb{R})$. 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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Mukul Dwivedi, Tanmay Sarkar. 2023-12-22. Convergence of a conservative Crank-Nicolson finite difference scheme for the KdV equation with smooth and non-smooth initial data. https://arxiv.org/abs/2312.14454

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