Tau Method

[Andrew Cook]

Hi all,

I recently started trying to replicate results of a paper using Dedalus. To do this, I have to solve a problem with Dirichlet boundary conditions in y. The highest order derivative with respect to y of a field in the problem is a third order derivative. I only have 2 boundary conditions for each such field, but from what I have read in the documentation I need 3 tau terms and 3 boundary condition for each field that has a third derivative with respect to y to use the tau method correctly in this case. I wasn't sure how to add a boundary condition, so I have tried estimating a solution where the third derivative terms are 0. This has caused my solution to not agree with the results of the paper.

I wanted to know how I should resolve this issue. Assuming my understanding of the restrictions of the tau method are correct, I can either approximate the third derivatives of fields with respect to y in terms of state variables and/or constants or I can add another boundary condition for each field that has a third derivative with respect to y. What I want to know is:

1. Is there a good method to approximate derivatives in terms of the lower order derivatives in Dedalus? if not,
2. How would I choose another boundary condition for each field with a derivative term with respect to y that does not have boundary condition and tau field?
   • Specifically for my case, I already have Dirichlet boundary conditions at the extremes of the y basis. so, I thought I could say that the derivatives at the center of the y basis is 0, but this caused a singular matrix error. For reference, Ly is the span of the y basis. 
     • my boundary conditions were: "dy(field)(y=Ly/2)=0"

-Andrew