High Order Two-Derivative Runge-Kutta Methods with Optimized Dispersion and Dissipation Error

In this work we consider explicit Two-derivative Runge-Kutta methods of a specific type where the function f is evaluated only once at each step. New 7th order methods are presented with minimized dispersion and dissipation error. These are two methods with constant coefficients with 5 and 6 stages. Also, a modified phase-fitted, amplification-fitted method with frequency dependent coefficients and 5 stages is constructed based on the 7th order method of Chan and Tsai. The new methods are applied to 4 well known oscillatory problems and their performance is compared with the methods in that of Chan and Tsai.The numerical experiments show the efficiency of the derived methods.

Automated summary

In this work, explicit Two-derivative Runge-Kutta methods where the function f is evaluated only once at each step are considered. New 7th order methods with minimized dispersion and dissipation error are presented, and their performance on four well-known oscillatory problems is compared to existing methods. Numerical experiments demonstrate the efficiency of the derived methods.