I'm reading an article from this website:
My question is about several formulations in that paper:
The paper is about a wave equation and the use of a Galerkin method to discretize in space.
(1)
Page 4
why does the author use the fraction subscript? A convention?
(2)
Page 5
Why $(\hat{q_h}v^-)_{j+\frac{1}{2}}$? Since this term is derived from the boundery of $I_j$, why not $q_h^-$ directly?
(3) Most important, the energy estimate, page 6
see (2.8) which is derived from (2.4). Why did the author simply differentiate over q and u, instead of using the chain rule?
Any help would be appreciated.
For (1):
When using staggered grids it is common to denote the cell center with integer index and the cell faces shifted by 1/2. I think that this is a common notation.
For (2):
If you have $u_{tt}=q_x$ and apply the projection you get $\int u_{tt} v dx-\int q_x v dx=0$. Now apply partial integration to $\int_{I_j} q_xv dx$ yielding $\int_{I_j} q_x v dx = - \int_{I_j}qv_x dx + [qv]_{\partial I_j}$ where $\partial I_j$ is the boundary of the interval, i.e. $j-1/2$ and $j+1/2$. The fact, that $[qv]_{\partial I_j}$ is written as $\hat qv^+-\hat qv^-$ means exactly this.
For (3):
I agree with Jason Knapp. As you can see by the $dx$ of the integral, you project your residual onto the spatial domain $V_k$ which has no time dependency. Thus you can first take the derivative w.r.t time and set make a concrete choice of $w=q_h$.