The problem addressed in this paper is the accurate interpolation of numerical solutions of high order partial differential equations (PDE)s. This problem arises frequently, for example, with the static adaptive method of lines where a numerical solutions from a previous grid needs to be mapped accurately onto a new adapted grid. In most previous studies the mapping is performed using cubic splines interpolation but this can be insufficiently accurate for high order (PDE)s such as the Korteweg-de Vries equations. In this paper, we will consider three standard interpolating algorithms (based on the use of cubic splines, quintic splines and quintic B-splines) and two non-standard interpolating

algorithms based on Fornberg interpolation and on almost collocation. Accurate interpolation of a numerical solution is also frequently used in the assessment of the performance of the solution method. In such cases, accurate evaluation of the phase and amplitude errors and accurate integration of the invariants of motion of solitary waves are used as measures of the reliability of the underlying solution method of the PDE. Numerical integration using adaptive quadrature routines is not applicable in this case because it relies on the ability to automatically choose the location of the abscissas. The basic approach investigated in this study uses the exact integration of the interpolants. Several alternatives based on different choices for the interpolation technique will be investigated and applied to problems arising from several invariants of motion for the conservation of mass, momentum and energy for the Korteweg-de Vries Burgers equations. Each of the integration algorithms is implemented in Fortran and MAPLE and is available from the authors. Several numerical examples are presented to illustrate important features of the propagation and interactions of solitary waves of the Korteweg-de Vries Burgers equations, the breakup of a Gaussian pulse into solitary waves, and the development of an undular bore.