Two‐derivative Runge‐Kutta methods with optimal phase properties

Abstract

In this work, we consider two-derivative Runge-Kutta methods for the numerical integration of first-order differential equations with oscillatory solution.We construct methods with constant coefficients and special properties as minimum phase-lag and amplification errors with three and four stages. All methods constructed have fifth algebraic order. We also present methods with variable coefficients with zero phase-lag and amplification errors. In order to examine the efficiency of the new methods, we use four well-known oscillatory test problems. KEYWORDSamplification error, phase lag, two-derivative Runge-Kutta methods MSC CLASSIFICATION65L05; 65L06

A trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström pair to solve oscillating systems

“…Example The Orbital Problem in Kalogiratou et al 21 : $$$$ {\displaystyle \begin{array}{cc}\hfill {y}_1^{{\prime\prime} }& =-{y}_1+\frac{1}{1000}\cos (x),{y}_1(0)=1,{y}_1^{\prime }(0)=0,\hfill \\ {}\hfill {y}_2^{{\prime\prime} }& =-{y}_2+\frac{1}{1000}\sin (x),{y}_2(0)=0,{y}_2^{\prime }(0)=\frac{9995}{10000},x\in \left[0,10\right].\hfill \end{array}} $$$$ The exact solution is $$$$ {y}_1(x)=\cos (x)+\frac{1}{2000}\kern0.1em x\sin (x), $$$$ $$$$ {y}_2(x)=\sin (x)-\frac{1}{2000}\kern0.1em x\cos (x). $$$$ Now we consider …”

Development of an Efficient Diagonally Implicit Runge–Kutta–Nyström 5(4) Pair for Special Second Order IVPs

“…Definition 1. Given the method in ( 2)-( 4), an interval I = (− hs , 0) is said to be an interval of absolute stability if for all h ∈ I, it is |λ 1,2 | < 1, where λ 1,2 are the solutions of the equation in (19).…”

“…Definition 2. An interval (− hp , 0) corresponding to the RKN method in Equations ( 2)-( 4) is said to be an interval of periodicity if for every h ∈ (− hp , 0), |λ 1,2 | = 1, with λ 1 = λ 2 , where λ 1,2 are the roots of the equation in (19).…”

Exponentially fitted two-derivative DIRK methods for oscillatory differential equations

“…An advantage of these methods is that the number of algebraic order conditions is significantly reduced in comparison with the classical Runge-Kutta methods of the same order, thereby allowing the construction of high-order schemes with only a few stages. Following this, implicit TDRK methods were derived in [13,4], general linear TDRK methods were presented in [1,2,3,11], and recently TDRK methods with optimal phase properties were constructed in [32]. In the case if a good estimate of frequency is known in advance, one can further improve the numerical solution of these methods by incorporating the exponential fitting idea [22,8,29,15] in order to integrate exactly the system who solutions are linear combinations of e ˘iωx (ω is a prescribed frequency).…”

“…In particular, with s " 2 stages we obtain superconvergent schemes of order p " 2s and p " 2s `1, and with s " 3 stages, a method of order p " 2s is derived. This is a significant improvement since previous EFTDRK methods (either explicit or fully implicit) [12,32,4,13] only achieve orders p " 2s for s " 2 stages and p " 2s ´1 for s " 3 stages.…”