Convergence on error correction methods for solving initial value problems

Higher-order semi-explicit one-step error correction methods(ECM) for solving initial value problems are developed. ECM provides the excellent convergence O(h^2p+2)$O\left(h^{2p+2}\right)$ one wants to get without any iteration processes required by most implicit type methods. This is possible if one constructs a local approximation having a residual error O(h^p)$O\left(h^p\right)$ on each time step. As a practical example, we construct a local quadratic approximation. Further, it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second-order methods which can be improved into a semi-explicit type ECM of the order of accuracy 6$6$. The stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non-stiff problems. It should be remarked that the ECM approach developed here does not yield explicit methods, but semi-implicit methods of the Rosenbrock type. Both ECM and Rosenbrock’s methods require to solve a few linear systems at each integration step, but the ECM approach involves 2p+2$2p+2$ evaluations of the Jacobian matrix per integration step whereas the Rosenbrock method demands one evaluation only. However, it is much easier to get high order methods by using the ECM approach.