How to prove the strong formulation for Discontinuous Galerkin is conservative?

I've been trying to figure out how to prove that the following equation is conservative

$$\int_{D^k} \partial_t(u^k) v_j^k + \partial_x(f) v_j^k dx = [(f -f^*) v^k_j]_{x^k}^{x^{k+1}}$$

where $v_j^k$ is a test function for element $k$, $f$ is the flux and $f^*$ is the numerical flux.

Can anyone point me out to some reference?

• I don't think this would be regarded as conservative unless the right hand side is zero. Jan 11, 2015 at 21:09

What you've written is considered the "strong form" of DG. If you integrate by parts the spatial term and rearrange you get $$\int_{D^k} \left(\partial_t(u^k)v^k - f \partial_x(v^k)\right) dx = -[f^* v^k]_{x^k}^{x^{k+1}} = 0.$$ Taking $v = 1$ then gives $$\int_{D^k} \partial_t(u^k) dx = f^*(x^{k})-f^*(x^{k+1}).$$ This is a restatement of conservation - that change in the amount of $u^k$ over the element $D^k$ is equal to flux in minus flux out.
• You can just plug in $v^k = 1$ and then integrate by parts to get the same result as well. The "strong form" of DG is really unnecessary mathematically (hence we undo it to show conservation), but is convenient computationally. Jan 11, 2015 at 22:04