Balance-characteristic schemes with separated conservative and flux variables

The new approach is proposed for the development of difference methods with high resolution for the equation of convection with regard to diffusion. It based on the introduction of two different types of variables – “conservative” and “flux”, corresponding to the realization of the conservation law and correct calculation of characteristic region of influence respectively. The process of creation of new algorithms consists of two stages. At the first stage, linear uniform conservative difference schemes with improved dissipative and dispersion characteristics are constructed on the minimal computing stencil. At the second stage, conservative algorithm of minimal correction of the calculated values is used for the realization of the sufficient conditions of the principle of maximum. Explicit monotone algorithms were developed, that are stable in the case of the Courant number is less than unit and have the second order of accuracy on the smooth solutions. It is shown that new algorithms have noticeable advantages in comparison with the well-known TVD-schemes, based on the limitation of fluxes.