On convective terms approximation approach that corresponds to pure convection

Abstract

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.