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

Kalogiratou, Z.; Monovasilis, Th.; Simos, Theodore E.

doi:10.1002/mma.5487

Cited by 17 publications

(7 citation statements)

References 25 publications

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“…Kalogiratou et al used the first two simplifying assumptions:

A e = C, A C e + Â e = \frac{C^{2} e}{2} .

Then, the number of order conditions is reduced to 3, 5, and 9 for orders 3, 4, and 5. In addition to the quadrature order conditions, there is one condition for order 4:

b^{T} false(A C^{2} e + 2 Â C e false) + {\overset{b}{true}}^{T} C^{2} e = \frac{1}{12},

and three conditions for order 5:

b^{T} false(C A C^{2} e + 2 C Â C e false) + {\overset{b}{true}}^{T} false(C^{3} e + A C^{2} e + 2 Â C e false) = \frac{1}{15},

b^{T} false(A C^{3} e + 3 Â C^{2} e false) + {\overset{b}{true}}^{T} C^{3} e = \frac{1}{20},

b^{T} false(A^{2} C^{2}

…”

Section: Explicit Tdrk Methodsmentioning

confidence: 99%

“…It is common in the literature to modify a method with constant coefficients. Here, we modify the method given in Kalogiratou et al using the constant coefficients c i , a ij , and

{\overset{a}{true}}_{i j}

. We solve equations ϕ ( v )=0 and α ( v )=0 with the quadrature equations Q ( k ) up to k =4 for the coefficients b i and

{\overset{b}{true}}_{i}

, which are the following.…”

Section: Construction Of Explicit Tdrk Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Two‐derivative Runge‐Kutta methods with optimal phase properties

Kalogiratou

Monovasilis

Simos

2019

Math Methods in App Sciences

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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

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“…Kalogiratou et al used the first two simplifying assumptions:

A e = C, A C e + Â e = \frac{C^{2} e}{2} .

Then, the number of order conditions is reduced to 3, 5, and 9 for orders 3, 4, and 5. In addition to the quadrature order conditions, there is one condition for order 4:

b^{T} false(A C^{2} e + 2 Â C e false) + {\overset{b}{true}}^{T} C^{2} e = \frac{1}{12},

and three conditions for order 5:

b^{T} false(C A C^{2} e + 2 C Â C e false) + {\overset{b}{true}}^{T} false(C^{3} e + A C^{2} e + 2 Â C e false) = \frac{1}{15},

b^{T} false(A C^{3} e + 3 Â C^{2} e false) + {\overset{b}{true}}^{T} C^{3} e = \frac{1}{20},

b^{T} false(A^{2} C^{2}

…”

Section: Explicit Tdrk Methodsmentioning

confidence: 99%

“…It is common in the literature to modify a method with constant coefficients. Here, we modify the method given in Kalogiratou et al using the constant coefficients c i , a ij , and

{\overset{a}{true}}_{i j}

. We solve equations ϕ ( v )=0 and α ( v )=0 with the quadrature equations Q ( k ) up to k =4 for the coefficients b i and

{\overset{b}{true}}_{i}

, which are the following.…”

Section: Construction Of Explicit Tdrk Methodsmentioning

confidence: 99%

Two‐derivative Runge‐Kutta methods with optimal phase properties

Kalogiratou

Monovasilis

Simos

2019

Math Methods in App Sciences

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“…We note here that the assumption corresponding to (k = 2) for the RK methods A c = c 2 /2 cannot be satisfied for explicit methods. The authors in [15] presented a fifth-order method with four stages. They used the first two simplifying assumptions (k = 1 and k = 2):…”

Section: Algebraic Order Conditionsmentioning

confidence: 99%

“…The authors were the first to consider the general case; in [15], they presented methods of an algebraic order up to five, and in [14], they derived order conditions for trigonometrically fitted methods. As mentioned above, algebraic order conditions for the general ETDRK methods up to order five were given.…”

Section: Introductionmentioning

confidence: 99%

Construction of Two-Derivative Runge–Kutta Methods of Order Six

Kalogiratou,

Monovasilis

2023

Algorithms

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Two-Derivative Runge–Kutta methods have been proposed by Chan and Tsai in 2010 and order conditions up to the fifth order are given. In this work, for the first time, we derive order conditions for order six. Simplifying assumptions that reduce the number of order conditions are also given. The procedure for constructing sixth-order methods is presented. A specific method is derived in order to illustrate the procedure; this method is of the sixth algebraic order with a reduced phase-lag and amplification error. For numerical comparison, five well-known test problems have been solved using a seventh-order Two-Derivative Runge–Kutta method developed by Chan and Tsai and several Runge–Kutta methods of orders 6 and 8. Diagrams of the maximum absolute error vs. computation time show the efficiency of the new method.

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“…Another common test problem is the inhomogeneous equation:

y^{''} (x) = - 100 y (x) + 99 \sin (x), y (0) = 1, y^{'} (0) = 11,

with analytical solution:

y (x) = \cos (10 x) + \sin (10 x) + \sin (x) .

We integrated that problem

x \in ([], 0, 10 π)

as in Tsitouras and recorded the results in Table .…”

Section: Numerical Testsmentioning

confidence: 99%

Interpolants for sixth‐order Numerov‐type methods

Alolyan

Simos

Tsitouras

2019

Math Methods in App Sciences

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The classical four-stage family of explicit sixth-order Numerov-type method is considered. We provide two kinds of interpolants: (a) a three-step interpolation based on all available data at mesh points and (b) a local interpolant (ie, two steps) that is constructed after solving scaled equations of condition. These latter equations are explained and provided here. Applying these interpolants in a set of tests, we conclude that they produce global errors of the same magnitude with the underlying method. KEYWORDSy ′′ = f(x, y), explicit hybrid Numerov, interpolation MSC CLASSIFICATION 65L05; 65L06Math Meth Appl Sci. 2019;42:7349-7358.wileyonlinelibrary.com/journal/mma

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

Cited by 17 publications

References 25 publications

Two‐derivative Runge‐Kutta methods with optimal phase properties

Two‐derivative Runge‐Kutta methods with optimal phase properties

Construction of Two-Derivative Runge–Kutta Methods of Order Six

Interpolants for sixth‐order Numerov‐type methods

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