Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of the finite volume approximation to the solution u is an element of L-infinity (R-N x R) of the equation u(t) + div(vf(u)) = 0, where v is a vector function depending on time and space. A 'h(1/4)' error estimate for an initial value in BV(R-N) is shown for a large variety of finite volume monotonous flux schemes, with an explicit or implicit time discretization. For this purpose, the error estimate is given for the general setting of approximate entropy solutions, where the error is expressed in terms of measures in R-N and R-N x R. The study of the implicit schemes involves the study of the existence and uniqueness of the approximate solution. The cases where an 'h(1/2)' error estimate can be achieved are also discussed.