Thesis Seminar: Lagrangian Averaging, Nonlinear Waves, and Shock Regularization

Harish Bhat
California Institute of Technology
Control and Dynamical Systems

Taking a geometric point of view, we derive new models for mean compressible flow as well as new model equations for shock waves. The models for mean compressible flow are derived by averaging the Lagrangian that gives the compressible Euler equation. We analyze the dynamics of a one-dimensional version of these models, and find a family of traveling wave solutions. Based on this analysis, we devise criteria that a shock-capturing equation should satisfy. Applying these criteria, we write a Hamiltonian, non-local regularization of the inviscid Burgers equation. For this Hamiltonian PDE, we analyze both the traveling wave solutions and initial-value problem. We show that as the regularization parameter tends to zero, solutions of the regularized equation converge strongly to solutions of the inviscid Burgers equation. Numerical testing shows that the limit is entropic. Along the way, we explore the relationship between new models and old ones.