A combined space discrete algorithm with a Taylor series by time for CFD
  The first order by time partial differential equations are used as models in applications such as fluid flow, heat transfer, solid deformation, electromagnetic waves, and others. In this paper we propose the new numerical method to solve a class of initial-boundary value problems for the PDEs using one of the known space discrete numerical schemes and a Taylor series expansion by time. Normally a second order discretization by space is applied while a first order by time is satisfactory. Nevertheless, in a number of different problems, discretization both by temporal and by spatial variables is needed of highest orders, which complicates numerical solution, in some cases dramatically. Therefore it is difficult to apply the same numerical methods for the solution of some PDE arrays if their parameters are varying in a wide range so that in some of them different numerical schemes by time fit the best for precise numerical solution. The Taylor series based solution strategy for the non-stationary PDEs in CFD simulations has been proposed here that attempts to optimise the computation time and fidelity of the numerical solution. 
Ivan Kazachkov