Dissipative-dispersive properties of one projective method for numerical solution of the advection equation

In this paper, we investigate the dissipative-dispersive properties of projection- and of the characteristic method of the third order of approximation for numerical solution of the advection equations. This scheme is called Cubic Polynomial Projection, and it is constructed by a grid-based scheme by the characteristic method using Hermite interpolation. The properties of this scheme are compared with similar properties of the Cubic Interpolation Polynomial scheme, widely used in computational practice and also based on Hermite interpolation. Both schemes belong to the class of characteristic schemes, which is important for particle transport problems and explicit consideration of the exponential dependence of the solution on the optical thickness. Instead of the traditional interpolation closure characteristic of the Cubic Interpolation Polynomial scheme, the Cubic Polynomial Projection scheme uses orthoprojector closure. This allows to transfer this scheme to unstructured tetrahedral meshes and solves the problem of coplanarity of the characteristic of one of the faces of the cell, but doubles the required memory resources in the simplest one-dimensional case. The paper shows that projective closure significantly improves the already quite good dissipative-dispersive properties of the Cubic Interpolation Polynomial scheme, significantly approaching them to the dissipative-dispersive properties of the exact solution of the advection equation. These conclusions are confirmed by numerical calculations.