如何在广义加权剩余方法（Chebyshev Galerkin Modal形式）中施加边界条件
我正在努力将广义加权剩余方法与RK4相结合。 GWRM零件将PDE分解到光谱空间，在该空间中，未知数是Chebyshev系数A_K。但是，我很难看到如何在这种情况下包括边界条件。在其他光谱方法中，包括物理网格，因此可以明确设置边界条件或包含在Chebyshev分化矩阵中。另一方面，我唯一的信息是边界处的解决方案之和，但边界取决于整个解决方案。因此，在每个RK4步骤中，边界永远不会明确设置。
这是我要解决的颂歌的简短派生。有人对如何包括边界条件有任何想法吗？
请记住 a ， b 和 c 都是向量。 Prime表示第一和术语除以2。PS
所得的方程是可以通过RK4离散化的ODE。
这是我目前对如何实现BC的理解，但是在每次步骤中，解决方案都越来越远离真实的边界条件。
最高模式K和K-1处的Chebyshev系数可以代替为边界方程式，
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,
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答案（75％确定吗？）是因为在光谱空间中没有明确的边界条件，因此无法明确的时间集成方案。基本功能必须满足边界条件，或者需要明确设置边界条件。
为了使用GWRM来求解PDE，您是否需要在光谱分解中包括时间域并求解一组线性/非线性代数方程，或者使用隐式时间集成方案（例如向后欧拉或隐式RK4）。
隐式方法起作用而不是明确方法的原因是，在隐式方法中，下一步步骤的chebyshev系数出现在公式的两侧。因此，您可以将最高模式替换为边界条件并迭代，直到Chebyshev系数的下一步满足PDE和边界条件。
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.