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 hyperbolic equation u t (x; t) + divF (x; t; u(x; t)) = 0 with the initial condition u 0 2 L 1 (IR N ). Passing to the limit in these schemes and using a technique of regularization, we prove existence and uniqueness of the entropy solution u 2 L 1 (IR N \Theta IR + ) and the convergence of the finite volume approximation to the entropy solution in L p loc (IR N \Theta IR + ), 1 p +1. Furthermore, if u 0 2 BV loc (IR N ), we show that u 2 BV loc (IR N \Theta IR + ) which leads to an "h 1 4 " error estimate between the approximate and the entropy solutions (where h defines the size of the mesh). 1 Introduction 1.1 Presentation of the problem We consider here the following nonlinear hyperbolic equation with some initial condition : ( u t (x; t) + div(F (x; t; u(x; t))) = 0; 8x 2 IR N ; 8t 2 IR + ; u(x; 0) = u 0 (x); 8x 2 IR N (1) where F : IR N \Theta IR + \Theta IR ! IR N ; N 1 (x; t; s..