Abstract
We present a relaxation semilinear system of conservation laws with source which approximates nonlinear parabolic equations with initial and boundary conditions. The system can be interpreted as a BGK (Bhatnagar, Gross, Krook) model with a finite number of velocities. We prove the well‐posedness of the model, a priori estimates and we obtain the convergence towards the solution of the parabolic problem. Moreover we prove a similar result for a weakly degenerate problem.