Mingyu Yu L&S Math & Physical Sciences
Numerical Study of Shock Formation Based on the Self-Similar Approach
Shock waves occur when there is a sudden change in the flow velocity, pressure, or density in a compressible medium (such as a gas). Mathematically, shock waves are types of singularity formed in the solutions to nonlinear PDEs, where the spatial gradient of the solution becomes infinite in finite time. Although significant progress has been made in developing numerical schemes for capturing shock waves in conservation laws, recent mathematical advances can help formulate novel numerical schemes. This project aims to create a robust numerical scheme to understand the formation of shock waves based on a self-similar approach. We apply this self-similar scheme to simulate the formation of shocks in Burgers’ equation and extend it to include other physical effects. We are particularly interested in two common phenomena: dispersion and dissipation. We study the stability of the formation of singularities from a numerical viewpoint, and we attempt to find scenarios in which the singularity formation is stable in the self-similar coordinates.