Accelerating convergence for backward Euler and trapezoid time discretization schemes

In this study, we introduce two algorithms to numerically solve any initial value problem (IVP). These algorithms depend on time relaxation model (TRM) which is obtained adding a time relaxation term into IVP. Discretizing TRM by using backward Euler (BE) method gives the first algorithm. Similarly, the second algorithm is followed by using trapezoid (TR) time stepping scheme . Under some conditions, the first algorithm increases the order of convergence from one to two and the second one increases the order from two to three. Thus, more accurate results can be obtained. To verify the accuracy of the methods, they are applied to some numerical examples. Numerical results overlap with the theoretical results.