Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate

In this paper, we study some finite volume schemes for the nonlinear hyperb olic equation u(t)(x, t) + divF(x, t, u(x, t)) = 0 with the initial conditi on u(o) is an element of L-infinity(R-N) Passing to the limit in these sche mes, we Drove the existence of an entropy solution u. is an element of L-in finity(R-N x R+). Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in L-loc(P)(R-N x R+), 1 less than or equal to p less than or equal to +infinity. Furthermore, if u(o) is an element of L-infinity boolean AND BVloc(R-N), we show that u is an element of BVloc(R-N X R+)1 which leads to an "h(1/4)" error estimate b etween the approximate and the entropy solutions (where h defines the size of the mesh).