### Godunov-Type Approximation for a General Resonant Balance Law with Large Data

Debora Amadori, Laurent Gosse, and Graziano Guerra

Abstract: We consider the Cauchy problem for the $2\times 2$ nonstrictly hyperbolic system

 at=0, (a,u)(0,\cdot)=(ao,uo). ut+f(a,u)x - g(a,u)ax =0,
For possibly large, discontinuous and resonant data, the generalized solution to the Riemann problem is introduced, interaction estimates are carried out using a new change of variables and the convergence of Godunov approximations is shown. Uniqueness is addressed relying on a suitable extension of Kru\v zkov's techniques.

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