Improved Rosenbrock method with error estimator and Jacobian approximation using complex step

This paper proposes an A-stable one-stage Rosenbrock method for the solution of Ordinary Differential Equations (ODEs). In this method, Jacobians are approximated via complex step finite differences. An asymptotically accurate estimator of the truncation error is also provided. This error estimator can be employed to control step sizes and to perform extrapolation, which increases the accuracy of the method and yields L-stability. Numerical experiments are conducted to assess the performance of the proposed method. ODE solvers and several stiff ODE problems from the current literature are employed as references during experiments. Experimental results reveal that the proposed method exhibits superior performance with respect to the other compared methods, especially for crude error tolerances.