Research Projects

The newly developed Godunov-type schemes for the pressureless gas dynamics equations by the author maintain the positivity property (vacuum state) very well while simultaneously capturing delta shocks with the maximum possible strengths, as it is one of the least diffusive schemes in the history of numerical methods for the pressureless gas dynamics systems. In addition to the above, the scheme is compact, simple to implement, and robust. By incorporating gravity terms, the given system could be easily extended to shallow water equations (SWE) systems. It is common practice in the research community to often use stable but very diffusive local lax-Freidrichs (LLF) method to investigate the numerical solutions of SWE systems. Since we successfully developed a highly accurate Godunov-type scheme for a system that could be treated as a convective part of the SWE system, it is natural to ask whether we can extend the given scheme to construct a stable, robust and accurate numerical scheme for systems of shallow water equations. Thus, more accurate yet robust numerical schemes for SWE systems may encourage researchers to use such schemes in their research studies. Likewise, Shear Shallow Water Equations (SSWE) model can be viewed as an extension of the SWE systems. In this model, shear effects are also taken into account. This system loses its strict hyperbolicity and becomes weakly hyperbolic in the flow field. Because of this reason, developing upwind schemes based on eigenvectors for the SSWE model is a challenging task. We want to approach this problem along the lines of the author's previous work in which the Jordan Canonical Forms were used to construct the Flux Difference Splitting (FDS) based upwind scheme for compressible Euler equations of the gas dynamics. Developing such a scheme for the SSWE model would be the first of its kind in the field of research.