Well-Balanced Schemes for the Initial Boundary Value Problem for 1D   Scaler Conservation Laws
  We consider well-balanced schemes for the following 1D scalar conservation law with source term: d_t u + d_x f(u) + z'(x) b(u) = 0. More precisely, we are interested in the numerical approximation of the initial boundary value problem for this equation. While our main concern is a convergence result, we also have to extend Otto's notion of entropy solutions to conservation laws with a source term. To obtain uniqueness, we show that a generalization, the so-called entropy process solution, is unique and coincides with the entropy solution. If the initial and boundary data are essentially bounded, we can establish convergence to the entropy solution. Showing that the numerical solutions are bounded we can extract a weak*-convergent subsequence. Identifying its limit as an entropy process solution requires some effort as we cannot use Kruzkov-type entropy pairs here. We restrict ourselves to the Engquist-Osher flux and identify the numerical entropy flux for an arbitrary entropy pair. By the uniqueness result, the scheme then approximates the entropy solution and a result by Vovelle then guarantees that the convergence is strong in L^p for finite p. 
M. Nolte, D. Kroener