Discussed time stepping for aeroacoustics with Antony Jameson at Stanford. Also reviewed a couple of his group’s papers on high order flux reconstruction schemes.

- Vincent, Castonguay, Jameson
- JCP 2011

Investigate a 1-parameter family of stable flux reconstruction methods suggested by earlier work. For certain parameter values you get DG or SD schemes. Some values admit spurious modes. The size of the largest stable step size and the order of accuracy are determined as a function of the parameter (c). Nonlinear 2D experimental results are predicted relatively well by the 1D von Neumann analysis.

Section 4 is a nice description of how to do von Neumann analysis for FE methods.

- Jameson, Vincent, Castonguay
- J. Sci. Comput. 2011

They look at energy stability in a very general way. Nonlinear stability depends on solution point locations, and on the accuracy of the determination of the transformed flux.

Time integration ideas that could be useful for aeronautics simulations:

- Optimization of stability regions
- Multirate time stepping. Some work has been done in Nonuniform time-step Runge–Kutta discontinuous Galerkin method for Computational Aeroacoustics.
- Large time step methods. See A class of large time step Godunov schemes for hyperbolic conservation laws and applications.

Large time step methods might work very well as the coarse propagator for parareal-type algorithms.