We present a high order discontinuous Galerkin (DG) method for the numerical solution of systems of delay differential equations (DDEs). The method is based on Legendre orthogonal polynomials of high degree k (typically k=10) in each subinterval and is a generalisation to DDEs of a similar method which proved to be very efficient for ordinary differential equations (ODEs). We show how the error can be estimated allowing to control the size of the time step. We also propose a particularly efficient quasi Newton method for the solution of the resulting non linear systems based on a very accurate approximation of the Jacobian matrix that is very easy to implement and with excellent convergence properties. The method is then applied to very stiff systems of DDEs.
