Revised trigonometrically fitted two-step hybrid methods with equation dependent coefficients for highly oscillatory problems

For the numerical integration of highly oscillatory problems, revised trigonometrically fitted two-step hybrid methods (RTFTSH) with equation dependent coefficients are considered. The local truncation errors, stability and phase properties of the new method are analyzed. A feature of the new type of the methods is that the errors in the internal stages are assumed to contribute to the accuracy of the update. A new revised method RTFTSH4 of algebraic order four and phase-lag order four is derived. Numerical experiments are reported to show that the new method RTFTSH4 is much more efficient and robust than the standard fourth order method STFTSH4.