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New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients
Author(s) -
Kalogiratou Zacharoula,
Monovasilis Theodore,
Simos Theodore E.
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5487
Subject(s) - mathematics , runge–kutta methods , derivative (finance) , work (physics) , order (exchange) , constant (computer programming) , variable (mathematics) , constant coefficients , second derivative , numerical analysis , mathematical analysis , computer science , thermodynamics , financial economics , economics , programming language , physics , finance
Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y ′ = f ( y ) that include the second derivative y ′ ′ = g ( y ) = f ′ ( y ) f ( y ) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.