A family of difference schemes solving the Cauchy problem for quasi-linear equations is studied. This family contains well-known schemes such as the decentered, Lax, Godounov or Lax-Wendroff schemes. Two conditions are given, the first assures the convergence to a weak solution and the second, more restrictive, implies the convergence to the solution in Kruzkov's sense, which satisfies an entropy condition that guarantees uniqueness. Some counterexamples are proposed to show the necessity of such conditions. The purpose of this study is the numerical solution of the Cauchy problem (1) ut + f(u)x = ? if (x, t) EE R x I O, T [ (2) u(x, O) = uo(x) if x E R, where uo E L?(R), with locally bounded variation on R, f E C1(R), and T > 0 are given. Section 1 recalls some theoretical results of existence and mainly of uniqueness for problem (1), (2), more particularly Oleinik's and Kruzkov's results. Section 2 is devoted to proofs of convergence for a family of numerical schemes; then Section 3 deals with various applications concerning some well-known numerical schemes (Lax, Godounov, Lax-Wendroff schemes, decentered scheme). 1. Since f is nonlinear, a classical solution u of (1), (2) may offer singularities after some value of t, even when uo is very regular. With a more general definition of the solution, we can extend u beyond this value of t. The notion of a weak solution represents one of these generalizations, but does not assure the uniqueness of the extension. These singularities of the solution make needless any hypothesis of regularity on the initial value uo. DEFINITION 1. U is a weak solution of (1), (2) when u E L (R x ]0 , T[), and (3) R XI O,T[ {utu + Oxf(u)} dx dt + R ?(X, O)uo(x) dx = 0, for all functions / twice continuously differentiable and with compact support on R 0 , T E() "2 (R x I O, T [) By multiplying (1) by 0 and integrating by parts, we obtain (3). The existence of a weak solution can be proved by the vanishing viscosity method (parameter e > 0) from a quasi-linear problem of parabolic type (4) (u) t + f(u,)x = (u,)x x if (x, t) E R x ]0, T[, u'(x, 0) = uo(x) if x E R. Received May 27, 1975. AMS (MOS) subject classifications (1970). Primary 35F25, 65M10. Copyright