Riemann Problem and Wave Interactions 

In the study of hyperbolic systems of conservation laws, particularly in multi-dimensions, understanding wave dynamics is both mathematically rich and physically significant. These systems, governed by hyperbolic partial differential equations, model a broad array of real-world phenomena from fluid dynamics and gas flows to traffic patterns and astrophysical processes. Central to this theory is the Riemann problem, which helps explain how discontinuities such as shocks, rarefactions, contact discontinuities, and even non-classical delta shocks evolve and interact over time.

In my research, I focus on analyzing such wave interactions using generalized characteristic analysis, combining with high-order stable numerical methods such as the second-order LLF scheme and the semi-discrete central upwind scheme.

• In my first paper, I investigated a two-dimensional three-state Riemann problem arising in a reduced hyperbolic model for thin film flow of a perfectly soluble anti-surfactant solution. The study focused on the intricate wave interactions involving rarefaction waves, shock waves, contact discontinuities, and delta shock waves.

• In my second work, I extended this analysis to the pressureless Euler system, investigating the intricate dynamics arising from the interaction of delta shocks and contact discontinuities in two dimensions. A key feature of the solution is the emergence of delta shock waves, characterized by Dirac delta distributions in both the density and internal energy. Some of these exhibit intriguing phenomena similar to Mach reflection and the emergence of a vacuum region in the solution.