Convergence of the fractional step method for a \(2\times 2\) nonstrictly hyperbolic system of conservation laws (Q1916715)

The author studies the existence of solutions of the Cauchy problem for a class of nonstrictly hyperbolic \(2\times 2\) systems of conservation laws with nonhomogeneous terms and an isolated umbilic point. Using compensated compactness, convergence to a weak solution is established for approximate solutions generated by fractional step versions of the Lax-Friedrichs method and the Godunov method. A technical innovation is the use of weak entropy/entropy flux pairs that vanish on a half plane. The analysis owes much to the pioneering work of P. T. Kan on adapting the method of compensated compactness to systems with quadratic fluxes in the class considered in this paper.