Fourth derivative singularly P-stable method for the numerical solution of the Schrödinger equation

In this paper, we construct a method with eight steps that belongs to the family of Obrechkoff methods. Due to the explicit nature of the new method, not only does it not require another method as predictor, but it can also be considered as a suitable predictive technique to be used with implicit methods. Periodicity and error terms are studied when applied to solve the radial Schrödinger equation, considering different energy levels. We show its advantages in terms of accuracy, consistency, and convergence in comparison with other methods of the same order appearing in the literature.

A Two-parameter Family of Exponentially-fitted Obrechkoff Methods for Second-order Boundary Value Problems

Generally, classical numerical methods may not be well suited for problems with oscillatory or periodic behaviour. To overcome this deficiency, they are modified using a technique called exponential fittings. The modification makes it possible to construct new methods suitable for the efficient integration of oscillatory or periodic problems from classical ones.In this work, a two--parameter family of exponentially--fitted Obrechkoff methods for approaching problems that exhibit oscillatory or periodic behaviour is constructed. The construction is based on a six-step flowchart described in [13]. Unlike the single--frequency method in [21], the constructed methods depend upon two frequencies which can be tuned to solve the problem at hand more accurately. The leading term of the local truncation error of the new family of method can also be easily obtained from the given general expression. The efficiency of the new methods is demonstrated on some numerical examples. This work is related to [20,21] and provides extension to the results obtained in [21]

A New Symmetric p-stable Obrechkoff Method with Optimal Phase lag for Oscillatory Problems

In this paper, we derive a class of symmetric p-stable Obrechkoff methods via Padé approximation approach (PAA) for the numerical solution of special second order initial value problems (IVPs) in ordinary differential equations (ODEs). We investigate periodicity analysis on the proposed scheme to verify p-stability property. The new algorithms possess minimum phase-lag error which shows that they can accurately solve oscillatory problems. Reports on several numerical experiments are provided to illustrate the accuracy of the method.