An optimized two-step hybrid block method for solving general second order initial-value problems

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.

“…Since then, several authors constructed EF/TF RK methods,() EF/TF RKN methods,() and two‐step hybrid methods. () Here, we will construct modified methods following both approaches.…”

Section: Construction Of the New Methodsmentioning

confidence: 99%

“…14 [18][19][20][21][22] and two-step hybrid methods. 11,23 Here, we will construct modified methods following both approaches.…”

Section: Methods With Variable Coefficientsmentioning

confidence: 99%

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Kalogiratou

Monovasilis

Simos

2019

“…Then, we choose the two body problem [48][49][50] with eccentricity 0.1, , 0] T . The corresponding results can be found in Table 4.…”

Section: Two Body Problemmentioning

confidence: 99%

Explicit hybrid six–step, sixth order, fully symmetric methods for solving y ″ = f (x,y)

Fang

Liu

Hsu

et al. 2019

A family of explicit, fully symmetric, sixth order, six‐step methods for the numerical solution of y′′ = f(x,y) is studied. This family wastes two function evaluations per step and can be derived through interpolation techniques. An interval of periodicity is possessed and the phase lag is of high order. Numerical instabilities usually present in such type of multistep methods were circumvented. We conclude with extended numerical tests over a set of problems justifying our effort of dealing with the new methods.

“…Problem 2. Inhomogeneous equation: It has the form, 47,48 x ′′ = −100x(t) + 99 sin(t), x(0) = 1, x ′ (0) = 11, with analytical solution, 49,50 x(t) = cos(10t) + sin(10t) + sin(t).…”

Section: Numerical Performancesmentioning

confidence: 99%

Phase‐fitted, six‐step methods for solving x′′ = f(t,x)

Liu

Hsu

Simos

et al. 2019

Self Cite

An explicit, six‐step method of sixth order is presented and tuned for the numerical solution of x′′ = f(t,x). This method is explicit, hybrid, and uses two function evaluations (stages) per step. Its coefficients are varied and depend on the step size. This variance comes from the demand of the method to nullify the phase errors produced when solving the standard simple oscillator. The first and second derivative of this error vanish also. Numerical tests in a set of relevant problems illustrate the efficiency of the newly derived method.