Speaker
Description
Computer simulations of the solution to conservation laws are important for the analysis of fluid flows, which is in turn used in the design of aircraft and spacecraft. Perhaps the most famous of these conservation laws are the Navier-Stokes equations, which describe the flow of most real fluids. If one considers an inviscid fluid, the Navier-Stokes equations reduce to the Euler equations, which can be used to model flows with large Reynolds numbers (or very low viscosity).
Solutions to the Euler equations can, even if the initial conditions are smooth, develop discontinuities (shocks). Understanding and analyzing these shocks is critically important for the design of craft capable of supersonic travel. This talk will focus on the use of the use of Automatic Differentiation (AD) and the application of a calculus of variations for solutions to conservation laws developed by Bressan & Marson in 1995 to the solution to the 2D Euler equations. Starting from the 1-D theory, we will examine some of the most interesting difficulties in extending the new calculus to a 2-D conservation law, and detail the use of:
- A local pseudoinversion technique for computing the sensitivity of the shock position to initial conditions
- Computing generalized tangent vectors using the shock sensitivities from local pseudoinversion
However, the extension to 2 space dimensions doesn't come "for free". We will also discuss some of the difficulties with the Bressan calculus, using AD to accumulate derivatives of a time-stepping method, and combining a shock-capturing finite volume method with the Bressan calclus.