Finite volume schemes for nonhomogeneous scalar conservation laws: error estimate

Summary.

In this paper, we study finite volume schemes for the nonhomogeneous scalar conservation law $u_t +{\rm div} F(x,t,u)=q(x,t,u)$ with initial condition $u(x,0)=\u0(x)$. The source term may be either stiff or nonstiff. In both cases, we prove error estimates between the approximate solution given by a finite volume scheme (the scheme is totally explicit in the nonstiff case, semi-implicit in the stiff case) and the entropy solution. The order of these estimates is $h^{\frac{1}{4}}$ in space-time $L^1$-norm (h denotes the size of the mesh). Furthermore, the error estimate does not depend on the stiffness of the source term in the stiff case.