The problem to calculate initial fields of concentration of a substance at every time step is inherent to computer calculations of the substance advection. The problem appears due to that at every new time step physical boundaries as a rule do not coincide with lines of the adopted calculation net. The lack of coincidence affects tremendously the accuracy of calculations. It becomes crucial in the case of 3D computations, when the distortion of the spatial net employed for computer simulation of 3D problems of fluid dynamics performed in the framework of Lagrange variables, may become so large that a subsequent time step gives values of physical quantities with extremely large errors so that further calculations become impossible. On account of this reason one is forced permanently to reconstruct the spatial net and correspondingly recalculate physical values from the old net to the new one. In order to perform this procedure it is necessary to calculate intersection volumes of cells of new and old nets. Therefore a necessity arises to invent a procedure allowing to calculate volumes of intersection of 3D bodies with various geometry so as to perform efficient 3D computer simulation of fluid dynamics. Presently there are available two approaches to compute intersection volumes of convex bodies arising in 3D computer sinmlations which are known to be conventional. One of them incorporates a fractionating technique of one of the intersection bodies onto a large number of smaller elementary ones similar to each other. The other method reduces the calculation of the intersection volume of cells to the analytical description of an intersection volume of two elementary bodies, for instance, tetrahedrons. The author has proposed his own new method of calculation of the intersection volume of 3D calculation cells based on the direct analytical description of the intersection volume of two arbitrary convex bodies. The method proposed was included in the algoritlun employed in the fluid dynanaics VNIIEF DF code for recalculation of physical values from the old net to the new one. It can easily be adopted to realize algorithms for calculations of intersection volumes of two arbitrary (not necessary convex) bodies with plain borders necessary for VNIIEF fluid dynamics code MEDUZA. The method was realized in FORTRAN-90 for the case of two convex bodies. It was employed to run test computations such as (a) initial fields of matter concentrations for a number of bodies with various geometries and (b) recalculation of flnid dynamics physical quantities from the old net to the new one. Results of calculations proved convincingly that the new method appeared to be from 10 to 30 times more efficient, that is less time consuming, in comparison with the above conventional methods and in many cases guaranties a much higher accuracy.