Consider the 1D advection equation in its strong and weak forms
$$\ u_t + a u_x = 0 $$ $$\ \int_{x_{j-0.5}}^{x_{j+0.5}} w u_t \ dx - a \int_{x_{j-0.5}}^{x_{j+0.5}} w_x u \ dx + a [w(x)\hat{u}]_{x_{j-0.5}} ^{x_{j+0.5}} = 0 $$
Having introduced the piecewise constant Lagrange polynomials or the first two Legendre Polynomials as our test functions, the system can be written as :
$$ \begin{bmatrix} \int (\phi_1)^2 dx \ \int \phi_1 \phi_2 dx \\ \int \phi_2 \phi_1 dx \ \int (\phi_2)^2 dx \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}_t - a \begin{bmatrix} \int (\phi_1)_x \phi_1 dx \ \int (\phi_1)_x \phi_2 dx \\ \int (\phi_2)_x \phi_1 dx \ \int (\phi_2)_x \phi_2 dx \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} + a \begin{bmatrix} \phi_1(x_{j-0.5}) \ \phi_1(x_{j+0.5}) \\ \phi_2(x_{j-0.5}) \ \phi_2(x_{j+0.5}) \end{bmatrix} \begin{bmatrix} -\hat{u}_{x_{j-0.5}} \\ \hat{u}_{x_{j+0.5}} \end{bmatrix} = 0 $$
When solving through the use of the Lagrange polynomial basis, the values $ u_1 $ and $ u_2 $ that are being solved for are the values of the function $ u(x,t) $ at the nodes. In the case of the Legendre Polynomials, what are we solving for? What is the physical significance of $ u_1 $ and $ u_2 $ and how do they relate to the function values at the nodes, cell average and flux ?
