Convective difference schemes and Hermite interpolation

A new method for numerically solving the convection equation is presented. The reduction of numerical errors is achieved by introducing an auxiliary variable and solving an auxiliary equation for the latter. The method is based on a Hermite interpolation of the convected quantity. Three explicit numerical versions of the method are investigated and compared with some of the most widely used schemes, and it is shown that the new method gives better physical results. Arguments are given that the usual mathematical series expansions do not work sufficiently well in that context. Instead of using linear stability analyses the quality of the method is investigated by numerical experiments.