  Recent decades are put lots of efforts to develop a higher-order scheme for
convective terms approximation that is stable and reliable. The idea presented
here is that approximation approach has to correspond to the physical
phenomenon described by approximated terms. Pure convection (advection) that is
described by convective terms is transporting a property along the streamline,
and the information propagation is unidirectional, i.e., transported property
depends on previous values along the streamline but does not depend on the next
ones. The proposed approach represents streamlines on mesh as discrete
streamlines and is called Discrete Stream(line) Method (DStreaM). A discrete
streamline here is represented as a narrow triangle with one vertex of the
approximated node and two others neighbor upstream nodes. Discrete streamlines
are orientated using local flow direction as skew upwind schemes. DStreaM
corresponds to pure convection. Here are considered standard test problems:
advection of a step profile, advection of a double-step profile, advection of a
sinusoidal profile, and Smith and Hutton problem. DStreaM solutions were
compared with upwind-first order scheme and second-order Total Variation
Diminishing (TVD) schemes with limiters Min-Mod, QUICK, and SUPERBEE solutions.
DStreaM demonstrated second-order accuracy and rapid convergence. Upwind and
DStreaM need 2 or 4 iterations to reach a final solution while TVD schemes need
from 15 to 93.5 more iterations. DStreaM approach looks promising for
calculation of convective-dominated problems because it approximates naturally
first derivatives and is straightforwardly applicable as a meshfree method or
on unstructured meshes.
