Penalty method for the KdV equation

Abstract

This article is the first part of a work in progress, related to the numerical simulation of hyperbolic equations with non-compatible data. The Korteweg–de Vries equations with non-homogeneous incompatible data are considered here. In this article we are able to prove the existence and uniqueness of solutions for the KdV equation despite the lack of regularity (compatibility) of the data. However, the lack of regularity is known to produce, for hyperbolic equations, large numerical errors which propagate within the whole domain. A method is proposed to replace the KdV equation with incompatible data by a system with compatible data using a penalty method. In this article we prove the existence and uniqueness of solutions of the exact and approximate problems and the convergence of the approximate solutions to the exact ones. The numerical simulations which justify the procedure will be presented elsewhere [N. Flyer, Z. Qin, and R. Temam, Numerical simulations of the penalty method for the KdV and Schrodinger equations, in preparation].

Keywords:

• Korteweg–de Vries equations
• hyperbolic system
• incompatible data

AMS Subject Classifications::

• 35Q53
• 35L57

Acknowledgements

This work was partially supported by the National Science Foundation under the grants NSF-DMS-0906440 and by the Research Fund of Indiana University.