A New Spatial Discretization Strategy of the Convective Flux Term for the Hyperbolic Conservation Laws

Abstract

In this work, a new spatial discretization scheme for flows governed by the hyperbolic conservation laws is proposed. The spatial discretization involves the concept of classical particle velocity upwinding (PVU) for the convective flux term in the hyperbolic conservation laws. The novelty of the approach lies in the use of the fluid particle velocity or the entropy wave speed at the cell interface to ascertain the upwind direction. The cell face convective fluxes are obtained from a first order or a second order upwind biased interpolation, depending on whether the cell under consideration lies in the vicinity of a discontinuity or in a region of steep gradients in the solution. The discontinuities or regions of steep gradients are detected by employing a smoothness indicator function as employed in some of the earlier studies. The proposed spatial discretization strategy has been combined with a two step, second order explicit time integration strategy for the application to the solution of the unsteady Euler/Navier-Stokes equations in the strong conservation form. Test cases involving two 1-D Riemann problems, three 2-D inviscid supersonic flow problems and a 2-D viscous supersonic flow problem, have been employed to establish the validity of the procedure and to assess the performance of the proposed strategy. The proposed PVU scheme performs quite favorably in comparison to conventional schemes. From the point of view of implementation, particularly in multidimensional scenarios, this strategy offers a good balance of accuracy and simplicity.