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CONVERGENCE OF A CONSERVATIVE CRANK-NICOLSON FINITE DIFFERENCE SCHEME FOR THE KDV EQUATION
  • MUKUL DWIVEDI,
  • Tanmay Sarkar
MUKUL DWIVEDI
Indian Institute of Technology Jammu

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Tanmay Sarkar
Indian Institute of Technology Jammu
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Abstract

In this paper, we study the stability and convergence of a conservative Crank-Nicolson finite difference scheme applied to the Korteweg-De Vries (KdV) equation endowed with initial data. We design a three-point average scheme associated to the convective term and the dispersion term is discretized by certain discrete operators along with the Crank-Nicolson scheme for the temporal discretization to establish that the proposed 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 ∈ H 3 ( R ) and to a weak solution in L 2 ( 0 , T ; L loc 2 ( R ) ) for non-smooth initial data u 0 ∈ L 2 ( R ) . Optimal convergence rates in both space and time for the devised scheme are derived. The theoretical results are justified through several numerical illustrations.
30 Jan 2024Submitted to Mathematical Methods in the Applied Sciences
30 Jan 2024Submission Checks Completed
30 Jan 2024Assigned to Editor
08 Feb 2024Review(s) Completed, Editorial Evaluation Pending
16 Feb 2024Reviewer(s) Assigned
10 Jul 20241st Revision Received
10 Jul 2024Submission Checks Completed
10 Jul 2024Assigned to Editor
10 Jul 2024Review(s) Completed, Editorial Evaluation Pending