Boundary conditions Chebyshev differentiation

Question

I was wondering if anyone has any experience dealing with boundaries when implementing chebyshev differentiation.

I am currently trying to implement a no slip boundary condition to solve the incompressible Navier Stokes equations in 3D, in order to ensure that flow is zero at the boundaries is it really just as simple as setting u(:,:,1) and u(:,:,N)=0 at every stage of computation (similarly for v and w) as is indicated in textbooks. This wouldn't seem to take into account how points next to the boundary are affected by there being zero flow at the boundaries and just seems far too simplistic an approach.

thanks to anyone who can help.

Answer 1

Dirichlet BCs are, by definition, a prescribed value at the boundary. If setting u(boundary) = 0 is unsettling to you, then consider the alternative of shrinking your domain so that you're only solving for the unknowns on the interior. Terms in the Navier-Stokes will reach to the boundary (where the velocity is known) but these velocities do not experience changes in momentum (they're purely kinematic).

One reason for including the boundaries themselves (and often ghost points) is to allow an easy change between Dirichlet BCs, where the boundary values are known, and Neumann BCs, where the values on the boundary must be solved for. The added point(s) are just a means to an end.

Answer 2

From my limited experience:

It takes into account algebraically but after doing arithmetic - plugging in zero nodal values (supposing that they are the unknowns in your approach) at boundaries - terms containing them vanish.

In general problem of applying Dirichlet boundary conditions the approach is the same as in any method where nodal values are unknown, and after discretization you get linear system from which you need to eliminate known/fixed DOF's.

Something that might be helpful:

https://code.google.com/p/another-chebpy/source/browse/p36-Laplace-nhBC.py