This paper considers computing accurate solutions on the interval [0, 1000] of ordinary differential equations. This includes implementation of high precision ode solvers and methods to verify the accuracy of the computed solution even for problems with chaotic behaviour. In this paper, we compute an accurate solution of the Lorenz system.We integrate the DOLFIN ODE solver with the GNU Multiple PrecisionLibrary (GMP) and are thus able to solve ODEs with arbitrary precision. We extend the ODE solver with general tools for a posteriori error analysis, including solving the linearized dual problem, and storing the primal solution and computing stability factors. In addition, we implement a number of optimizations in DOLFIN to make it possible to use methods with high order (∼ 200) with the solver.Using these tools we study the computability of the Lorenz system indetail and show that chaotic dynamical systems, like the Lorenz system, are indeed computable.