A numerical conception of entropy for quasi-linear equations

Abstract

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 Kružkov’s sense, which satisfies an entropy condition that guarantees uniqueness. Some counterexamples are proposed to show the necessity of such conditions.