Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws

摘要

We consider a general class of finite volume schemes on unstructured but quasiuniform meshes for first-order systems of hyperbolic balance laws on unstructured meshes. Provided the system is equipped with at least one entropy-entropy flux tuple and the associated Cauchy problem allows for a classical solution u we give conditions such that the finite volume approximation u(h) converges to u if the mesh parameter h tends to zero. In fact we prove an error estimate of the form parallel to u - u(h)parallel to (L)2 <= C root h, where C is independent of h. The proof relies on a stability result for classical solutions in the class of entropy solutions due to Dafermos [ Arch. Rational Mech. Anal., 94 ( 1979), pp. 373 - 389] and DiPerna [ Indiana Univ. Math. J., 28 ( 1979), pp. 137 - 188].