Entropy Stable Approximations of Nonlinear Conservation Laws

A central problem in computational fluid dynamics is the development of the numerical approximations for nonlinear hyperbolic conservation laws and related time-dependent problems governed by additional dissipative and dispersive forcing terms. Entropy stability serves as an essential guideline in the design of new computationally reliable numerical schemes. My dissertation research involves a systematic study of the novel entropy stable approximate methods of nonlinear conservation laws and application of those methods to solve one and two dimensional systems, e.g.) the Navier-Stoke equations, the shallow water equations, and more. We develop second-order difference schemes which avoid artificial numerical viscosity in the sense that their entropy dissipation is dictated solely by physical dissipation terms. The numerical results of 1D compressible Navier-Stokes equations equations provide us a remarkable evidence for different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts. Further implementation in 2D shallow water equations is realized dimension by dimension. These entropy-stable schemes also play a crucial role in the simulations of vanishing Leray-a smoothing model for Burgers equation. All the numerical experiments are implemented by a robust numerical package that offers a relatively simple, "black-box" solver for a wide variety of problems governed by nonlinear conservation laws.