Convergence of a finite difference method for the KdV and modified KdV\n equations with L² data

2012/02/06 by Paulo Amorim, Amorim, Paulo, Mário Figueira +1
Mathematics · #Advanced Mathematical Physics Problems #Differential Equations and Numerical Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1202.1232

Abstract

We prove strong convergence of a semi-discrete finite difference method for\nthe KdV and modified KdV equations. We extend existing results to non-smooth\ndata (namely, in L²), without size restrictions. Our approach uses a fourth\norder (in space) stabilization term and a special conservative discretization\nof the nonlinear term. Convergence follows from a smoothing effect and energy\nestimates. We illustrate our results with numerical experiments, including a\nnumerical investigation of an open problem related to uniqueness posed by Y.\nTsutsumi.\n

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