Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Yakubu, D. G.; Kumleng, G. M.; Markuš, Š.

doi:10.20454/jmmnm.2017.1349

Cited by 2 publications

(5 citation statements)

References 8 publications

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“…Again this points are also to give the zero stability as mentioned by (see, [24]) for accurate solution of the type of equation in (1). Expanding (20) fully, we get the type of continuous scheme in (15) as;…”

Section: The Seventh Order Second Derivative Implicit Block Hybrid Co...mentioning

confidence: 91%

“…These methods are constructed by introducing some polynomial nodes as interior collocation points and some evaluations of the second derivative g(x, y)in addition to the familiar evaluations of f (x, y). Although, many researchers have successfully derived and applied methods with second derivative evaluations (see, [10,11,12,13,14,15,16]) to mention but only a few examples. In this paper, we derive methods which yield very interesting results when applied to the type of equation in (1), give high order of accuracy within the standard interval of integration, and also behave essentially like one-step method.…”

Section: Introductionmentioning

confidence: 99%

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Two step hybrid methods for the integration of highly oscillatory systems of ODEs

Yakubu

Chollom

Momoh

et al. 2023

Preprint

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\noindent The use of collocation process for constructing numerical methods for solving ordinary differential equations has been attractive for stiff and problems with highly oscillatory solutions. In this paper, a class of block hybrid collocation methods has been developed which can efficiently solve stiff and highly oscillatory differential equations in block solution form. The block hybrid collocation methods are derived based on collocation at the polynomial nodes which are very effective for solving highly oscillatory systems. The block solution methods arising from the continuous formulation are discussed for various examples with applications. The methods are self-starting and produce dense output within the integration interval. The convergence of the derived methods is determined theoretically and asymptotic error constants are calculated. Improved performance over some known standard methods is achieved for a broad class of problems with oscillating solutions. Preliminary numerical calculations using our new methods clearly show improved performance, efficiency and effectiveness compared to some methods with strong algebraic stability properties. Efficiency curves of the solutions plotted, show rapid convergence of the proposed second-derivative block hybrid collocation methods. (2010) Mathematics Subject Classification:34M10,35B05,35B35,65L05

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Section: The Seventh Order Second Derivative Implicit Block Hybrid Co...mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Two step hybrid methods for the integration of highly oscillatory systems of ODEs

Yakubu

Chollom

Momoh

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…3.1.5. Region of Absolute Stability of the NMCFs (Yakubu et al [21], Lambert [37]) Finally, the stability of the NMCFs is examined and discussed using the procedure described in Yakubu et al [21] and Lambert [37].…”

Section: Convergence Of the Nmcfs (Henrici [39] Rufai Et Al [43])mentioning

confidence: 99%

“…Because of the aforementioned applications, various numerical techniques for solving various application problems, such as time-frequency analysis, signal delay, convection-diffusion equations, nonlinear approximation, and Monte Carlo simulation, arising in fluid dynamics problems have been developed. For instance, predictor-corrector techniques (Su et al [11], Awoyemi and Idowu [12], Iskandarov and Komartsova [13], Ashry et al [14], Asif [15]); Galerkin methods (Guo et al [16]); Haar wavelets methods (Aziz and Khan [17], Shiralashetti et al [18], Saparova et al [19]); Runge-Kutta methods (Takei and Iwata [20], Yakubu et al [21], Zhao and Huang [22]); Newtral network model (Mall and Chakraverty [23]); multigrid technique (Ghaffar et al [24], Ge [25], Gupta et al [26]); finite difference methods (Mulla et al [27]); and finite element methods (Harari and Hughes [28]).…”

Section: Introductionmentioning

confidence: 99%

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Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies

Omole,

Adeyefa,

Ayodele

et al. 2023

Axioms

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A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.

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Second derivative Runge-Kutta collocation methods based on Lobatto nodes for stiff systems

Cited by 2 publications

References 8 publications

Two step hybrid methods for the integration of highly oscillatory systems of ODEs

Two step hybrid methods for the integration of highly oscillatory systems of ODEs

Ninth-order Multistep Collocation Formulas for Solving Models of PDEs Arising in Fluid Dynamics: Design and Implementation Strategies

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