Numerical solution of first order initial value problem using 4-stage sixth order Gauss-Kronrod-Radau IIA method

Ying, Teh Yuan; Ya’acob, Norsuzila

doi:10.1063/1.4801121

Cited by 1 publication

(4 citation statements)

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“…with l j (τ) and v i (τ) defined by Equations ( 7) and (10), respectively. The right-hand side of Equation ( 8) is approximated as…”

Section: General Derivation Of the Methodsmentioning

confidence: 99%

“…By employing Equations ( 6) and (10), the right-hand side of Equation ( 8) finally becomes ∑ s j=1 f x n + c j h, y 0 + h ∑ s m=1 k m a jm q ij , i = 1, . .…”

Section: General Derivation Of the Methodsmentioning

confidence: 99%

“…In this case, the matrix P is a square matrix of size s × s. The nonzero determinant is a consequence of linearly independent equations (i.e., matrix rows), generated from test functions v i . These are, specifically, the Lagrange polynomials of degree s, Equation (10), with points ζ i as the roots of the Legendre polynomial. This is a sufficient condition for the matrix P to be invertible.…”

Section: Gauss-legendre and Lobatto Iiia Methods Families As Subclass...mentioning

confidence: 99%

“…Similarly, Vigo-Aguiar and Ramos [9], in addition to constructing an A-stable iRK method, proposed an embedding technique for changing the step size using the Runge-Kutta method and a linear multistep one. Teh and Yaacob [10] recently developed several implicit RK methods, two of them being Gauss-Kronrod methods with five stages and order 8 accuracy. These are based on five-point Gauss-Kronrod quadrature formula.…”

Section: Introductionmentioning

confidence: 99%

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Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Urevc

Halilovic̆

2021

Mathematics

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In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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“…with l j (τ) and v i (τ) defined by Equations ( 7) and (10), respectively. The right-hand side of Equation ( 8) is approximated as…”

Section: General Derivation Of the Methodsmentioning

confidence: 99%

“…By employing Equations ( 6) and (10), the right-hand side of Equation ( 8) finally becomes ∑ s j=1 f x n + c j h, y 0 + h ∑ s m=1 k m a jm q ij , i = 1, . .…”

Section: General Derivation Of the Methodsmentioning

confidence: 99%

Section: Gauss-legendre and Lobatto Iiia Methods Families As Subclass...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations