For scalar conservation laws in one space dimension with a flux function discontinuous in
space, there exist infinitely many classes of solutions which are L1 contractive.
Each class is characterized by a connection (A,B) which determines the interface entropy. For
solutions corresponding to a connection (A,B), there exists convergent numerical schemes
based on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes,
corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., used
widely in applications. In this paper we completely answer this question for more general
stable monotone schemes using a novel construction of interface flux function. Then from
the singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, we
prove the convergence of the schemes.