How to impose boundary conditions in the Generalized Weighted Residual Method (Chebyshev Galerkin modal form)

Question

I am working on combining a Generalized Weighted Residual Method with an RK4. The GWRM part decomposed the PDEs to the spectral space where the unknowns are Chebyshev coefficients a_k. However, I'm having difficulty seeing how the boundary conditions can be included in this case. In other Spectral methods, the physical grid is included and thus the boundary conditions can be set explicitly or included in the Chebyshev differentiation matrices. Here on the other hand the only information I have is the sum of the solution at the boundaries, but the boundary depends on the entire solution. So in each RK4 step the boundaries are never explicitly set.

Here is a short derivation of the ODE that I'm solving. Does anyone have any ideas on how the boundary conditions can be included?

Keep in mind A, b, and c are all vectors. Prime means first sum term is divided by 2.

P.s the resulting equations are ODEs which can be discretized with rk4.

This is my current understanding of how BCs are implemented but with each time step the solution gets further and further away from the true boundary conditions.

The Chebyshev coefficients at the highest modes K and K-1 can be substituted for boundary equations as such,

Answer 1

The answer (75% sure?) is that since there is no explicit boundary condition in the spectral space an explicit time integration scheme is not possible. Either the basis functions have to fulfill the boundary conditions or the boundary conditions need to be set explicitly.

In order to use the GWRM for solving PDEs, either you need to include the temporal domain in the spectral decomposition and solve a set of linear/nonlinear algebraic equations, or you use an implicit time integration scheme like Backward Euler or implicit RK4.

The reason the implicit methods work, and not the explicit method, is that in the implicit method the Chebyshev coefficients for the next time step appear on both sides of the equation. Thus you can substitute the highest modes for boundary conditions and iterate until the next step of Chebyshev coefficients satisfy the PDE and boundary conditions.