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A compactness framework and a convergence theorem for the LaxFriedrichs scheme and the Godunov scheme applied to the Cauchy problem for a 2x2 nonstrictly hyperbolic system of conservation laws are established. The existence of weak solutions is proved using the theory of compensated compactness of Tartar, Murat, DiPerna, and Serre.

Compactness framework and convergence of Lax-Friedrichs and Godunov schemes for a 2×2 nonstrictly hyperbolic system of conservation laws