kburns
commented
11 years ago Original comment by Benjamin Brown (Bitbucket: Benjamin Brown).
- set attachment to "1d_burgers_equation_chebyshev.png"
Burgers equation is now implemented in both Fourier and Chebyshev domains.
Early results look good; we're able to at least qualitatively match [see attached figure] fig 18 from Ascher et al. (1995). This test was done with nu=0.01, dt=1e-2, and nx=64. These are different choices than Ascher et al., except nu.
Original report (archived issue) by Benjamin Brown (Bitbucket: Benjamin Brown).
The original report had attachments: 1d_burgers_equation_chebyshev.png
Dedalus needs a time-step stability test suite.
We need an automated testing suite that can map out the stability domain for our different time-stepping algorithms. In particular, we should be able to create stability region maps like in Ascher et al. 1995 and Canuto et al. Appendix D, Figures D1 & D2.
In Dedalus, we can do this by solving a given linear equation set on a grid of variables, testing whether the amplitude (maybe integrated over the domain; L2 norm?) grows between successive timesteps.
Most previous work has been done semi-analytically and only addresses 2nd order finite difference or fourier spectra. An advantage over work done previously is that we can numerically map out this stability domain for chebyshev and legendre grids, where some of the answers are not well known and hueristic rules are generally applied.
We can also map this domain out for different equation sets. We should start with the simple 1-D advection-diffusion test, with
and constant advection speed
cand diffusion coefficientlambda, but we could eventually test the thermal mode equation, etc.This is straightforward to do for linear equation sets. Testing non-linear timestepping is more challenging, but we might look at Burgers equation (as in Ascher et al. 1995).
I can't assign this to me right now, but I'll take this on, as I need to understand why the CNAB3 timestepper is going unstable for an entropy-diffusion test.
--Ben