A Novel upwind method for genuine weakly hyperbolic systems and its application to Euler equations

Abstract: In this study, we examine construction of upwind methods to simulate genuine weakly hyperbolic (GWH) systems.  Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressure-less gas dynamics system, modified Burgers system and further modified Burgers system.  The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Difference Splitting (FDS), is to recover full set of LI eigenvectors. A novel  FDS-J solver is capable of capturing various shocks such as \(\delta\)-shocks, \(\delta´\) -shocks and \(\delta''\) -shocks accurately.  This strategy is further applied to those weakly hyperbolic subsystems which result on employing various convection-pressure splittings to the Euler flux function.  For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler flux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts.  Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems.