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Many physical phenomena can be modeled by partial differential equations, such as Euler equations and Navier–Stokes equations. However, most cases, we cannot find the exact solutions, hence numerical methods are needed to approximate them. In my work, I use discontinuous Galerkin methods (a kind of finite element method) and finite volume methods to solve hyperbolic and parabolic equations—especially for problems with strong singularities, such as delta-functions. Moreover, some of the physical bounds will be preserved, such as the positivity of the density, maximum-principle of the velocity etc. Some of the problems I am working on include compressible gas dynamics, miscible displacement in porous media, combustion, chemotaxis, radiative transfer equations. Some future works include numerical simulations in traffic flow, numerical cosmology, tsunami and hurricane prediction etc.
Yang, Yang, "Numerical methods for partial differential equations" (2016). TechTalks. 12.