Convergence of numerical schemes for short wave long wave interaction equations (Q2891099)

The authors consider, for their topic of wave interaction, the Cauchy problem of a simplified system going back to Benney: a nonlinear Schrödinger equation and a conservation law, both coupled through their nonlinear right-hand sides. The system is approximated by an infinite system of ODEs in time by using well known finite difference methods (here called finite volume schemes), employing, for the conservation law, flux functions with the usual properties (consistent, monotone etc.).NEWLINENEWLINEThe main body of the paper is devoted to a convergence proof of a subsequence of the discrete solution in an \(L^1_{loc}\) space, using their own former work [\textit{J. P. Dias, M. Figueira} and \textit{H. Frid}, Arch. Ration. Mech. Anal. 196, No. 3, 981--1010 (2010; Zbl 1203.35020)], and the compensated compactness method of Murat and Tatar. The paper shows a series of computing results in case the full discretization by a semi-implicit Crank-Nichelson scheme and the Lax-Friedrichs scheme are used (for linear and for nonlinear fluxes, and the full system with a piecewise constant function for the initial value of the conservation law). These results look well, however it is strange to see logarithms (of the errors) between 0.1 and 0.0001.