I am trying to check the order of total energy conservation of the collocation method (defined below) applied on the rotational nonlinear shallow water model, regardless of the spatial discretization. I will guide you with my attempt so far, but I am puzzled as to how to proceed further.
Suppose an ODE defined as: $$y'(t,y(t))=f(t,y(t)).$$
Following Costabile and Napoli (2011), the collocation method for an $n$-degree polynomial, is defined by:
$$y_n(t)=y(t_0)+\sum_{i=1}^n p_{n,i}f(t_i),$$ where $p_{n,i}=\int_{t_0}^t l_i(t') dt$ for $ i=1,...,n$, and $l_i$ is the lagrange polynomial.
Now, the rotational shallow water equation system is:
$$h' = -\nabla\cdot \mathbf uh$$ $$\mathbf u' = -\mathbf u\cdot\nabla\mathbf u+\omega\mathbf u^\perp-g\nabla(h+b)$$
where $h$, $\mathbf u$ is the scalar and vector fields, respectively, $b$ is the topography/bathymetry, and $u^\perp=\mathbf k\times\mathbf u$.
I am solely interested in the time discretization, therefore I am going to assume the RHS is exact. In that case, the collocation method for this system is:
$$h_n(t) = h(t_0)+\sum_{i=1}^np_{n,i} h'(t_i)$$ $$\mathbf u_n(t) = \mathbf u(t_0)+\sum_{i=1}^np_{n,i} \mathbf u'(t_i)$$
The total energy of this system is defined as:
$$E_T(t') = \frac{|\mathbf u(t')|^2}{2}h(t')+gh(t')\left(\frac{h(t')}{2}+b\right).$$
Applying the collocation method and giving some manipulation:
$$E_T(t)-E_T(t_0)= \delta_u h(t_0)\mathbf u(t_0)+\delta_h\frac{|\mathbf u(t_0)|^2}{2}+g\delta_h(h(t_0)+b)+\\\delta_u^2h(t_0)+\delta_h\frac{\delta_u'^2}{2} + g\frac{\delta_h'^2}{2},$$
where $\delta_f=\sum_{i=1}^n p_{n,i}f'(t_i)$.
So, I was wondering if there is a way to manipulate this equation further so that I can find the order of error in order to see the order of energy conservation. Maybe there is an upper bound that can be found?
Thank you very much for your time!
Side Note: If I were to apply the collocation method on the time derivative of the total energy itself:
$$E_T(t)-E_T(t_0)=\sum_{i=1}^n \mathbf u'(t_i)\mathbf u(t_i)h(t_i)+h'(t_i)\frac{|\mathbf u(t_i)|^2}{2}+gh'(t_i)(h(t_i)+b).$$
This equation is similar to the first three terms of the previous equation. If that were truly the case, then the order of error of energy would be the same as applying the collocation method on the total energy.