An analysis of numerical dispersion in approximations of the linear korteweg-de vries equation

Abstract

We consider a class of purely dispersive difference schemes for the linear Korteweg-deVries equation and analyze the pointwise behavior of the approximations when the initial data is a step function. Using a normal form analysis, we give a complete description of the approximation in a large region surrounding the leading front of the solution. We show in particular that in a fixed region around the front, numerical dispersion does not have a lasting effect on the approximation as the mesh size tends to zero. This contrasts sharply with the behavior of dispersive approximations of hyperbolic problems.

Keywords:

• Numerical dispersion
• Korteweg-deVries
• oscillatory integral
• normal form