We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. We construct an approximate solution to the problem via an elliptic approximation for which, under appropriate assumptions, we prove that the corresponding limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. We also provide numerical examples.
Mišur, M., Mitrovic, D., & Novak, A. (2016). On the Dirichlet-Neumann Boundary Problem for Scalar Conservation Laws. Mathematical Modelling and Analysis, 21(5), 685-698. https://doi.org/10.3846/13926292.2016.1214187
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