A penalty method for numerically handling dispersive equations with incompatible initial and boundary data

This article is the numerical counterpart of a theoretical work in progress Qin and Teman, Applicable Anal (2011), 1–19, related to the approximation of evolution hyperbolic equations with incompatible data. The Korteweg‐de Vries and Schrödinger equations with incompatible initial and boundary data are considered here. For hyperbolic equations, the lack of regularity (compatibility) is known to produce large numerical errors which propagate throughout the spatial domain, destroying convergence. In this article, we numerically test the effectiveness of the penalty‐based method proposed in Qin and Teman, Applicable Anal (2011), 1–19, which replaces the hyperbolic equations with incompatible data by a system with compatible data. We observe that convergence is increased. As explained in the text, in the case of the Schrödinger equation, the impact of incompatible (nonregular) data is most severe, and the authors are not aware of any other method that can handle such severe incompatible data. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011