An efficient Runge-Kutta (4,5) pair

Bogacki, P.; Shampine, L. F.

doi:10.1016/0898-1221(96)00141-1

Cited by 87 publications

(76 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The solution of ordinary differential equations arising in fluid flow were also solved using Matlab bvp4c based on Runge-Kutta method. The method is adaptive (adjusts) the numerical step size at every iteration Bogacki and Shampine [17]. The Runge-Kutta method is easy to use and more accurate than the traditional finite difference methods, Shampine [18].…”

Section: Introductionmentioning

confidence: 99%

Free Convection Fluid Flow from a Spinning Sphere with TemperatureDependent Physical Properties

Makanda¹

2017

Proceedings of the 4th International Conference of Fluid Flow, Heat and Mass Transfer (FFHMT'17)

View full text Add to dashboard Cite

-Natural convection from a spinning sphere with temperature dependent viscosity, thermal conductivity and viscous dissipation was studied. A unique system of non-similar partial differential equations was solved using the bivariate local-linearization method (BLLM). This method use Chebyshev spectral collocation method applied in both the η and ξ directions. Similar equations in the literature are normally solved by inaccurate time-consuming finite difference methods. This work introduces a robust method for solving partial differential equations arising in heat and mass transfer. The numerical method was validated by comparison to the results previously published in the literature. The method is fully described in this article and can be used as an alternative method in solving boundary value problems. This work also presents rarely reported results of the effect of selected parameters on spin-velocity profiles g(η).

show abstract

Section: Introductionmentioning

confidence: 99%

Free Convection Fluid Flow from a Spinning Sphere with TemperatureDependent Physical Properties

Makanda¹

2017

Proceedings of the 4th International Conference of Fluid Flow, Heat and Mass Transfer (FFHMT'17)

View full text Add to dashboard Cite

show abstract

“…Classical Runge-Kutta methods are very popular for solving (1). An s-stage RK method is defined by its Butcher tableau and given y n ≈ y(t n ) it takes an appropriate stepsize h to generate y n+1 ≈ y(t n + h) using…”

Section: Introductionmentioning

confidence: 99%

“…It is suitable for high accuracy of 10 −6 to 10 −12 . Also worthy of note is the more recent BS45 pair of Bogacki and Shampine [1], which uses s = 8 with FSAL and thus has the effective workload of a 7-stage method. It was reported in [1] to be more efficient than DOPRI5.…”

Section: Introductionmentioning

confidence: 99%

“…Also worthy of note is the more recent BS45 pair of Bogacki and Shampine [1], which uses s = 8 with FSAL and thus has the effective workload of a 7-stage method. It was reported in [1] to be more efficient than DOPRI5. Curiously though, Shampine and Reichelt [21] did not document why they left it out and instead retained DOPRI5 as their ode45 implementation in the MATLAB ODE suite.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Functionally fitted explicit pseudo two-step Runge–Kutta methods

Hoang

Sidje

2009

Applied Numerical Mathematics

View full text Add to dashboard Cite

“…The ode23 and ode45 solvers implement "Runge-Kutta formula of order 2 and 3" [21] and the "Dormand-Prince (4,5) pair" [28], respectively. The Dormand-prince (4,5) pair formula, which is an improved version of Fehlberg (4,5) pair [29], gives better results when compared with other Runge-Kutta schemes [30][31]. The ode113 solver implements the Adams' formulas of variable order in a Predict, Evaluate,…”

mentioning

confidence: 99%

Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model

Emagbetere

Oluwole

Salau

2017

BMSA

View full text Add to dashboard Cite

Abstract. Changing parameters of the Korakianitis and Shi heart valve model over a cardiac cycle has led to the investigation of appropriate numerical technique(s) for good speed and accuracy. Two sets of parameters were selected for the numerical test. For the seven MATLAB ODE solvers, the computed results, computational cost and execution time were observed for varied error tolerance and initial time steps. The results were evaluated with descriptive statistics; the Pearson correlation and ANOVA at 0.05 . The dependence of the computed result, accuracy of the method, computational cost and execution time of all the solvers, on relative tolerance and initial time steps were ascertained. Our findings provide important information that can be useful for selecting a MATLAB ODE solver suitable for differential equation with time varying parameters and changing stiffness properties. IntroductionThe Korakianitis and Shi (KS) heart valve model is a zero-dimensional, second order, nonlinear, Ordinary Differential Equation (ODE) that describes the dynamics of the human heart valve. The equation is embedded in the overall model of other parts of the entire cardiovascular system [1]. The equation may change from being non-stiff to stiff and vice versa, as its parameter changes within a heart cycle. An equation is said to be stiff if certain numerical schemes that are not absolutely stable cannot be used to seek an approximate solution to the problem [2]. Consequently, the conventional fourth order explicit Runge-Kutta method used to find a solution to the KS heart valve model in previous studies [1,[3][4] may not give accurate results for large step size, while smaller steps size may increase the round-off errors and increase computation time.On cannot exhaust the list of numerical techniques and their suitability for solving ODE as older ones are being modified and newer ones are developed. Butcher [5] reported a good number of these solvers developed over the 20 th century. Evaluations of ODE solvers have a long history [6-9] and earlier intentions were either to ascertain their effectiveness [6][7] or to categorize them as either stiff or non-stiff solvers [10][11][12][13]. Some researchers, however, reported the performance of the solving environment for some of these ODE solvers [14][15][16]. Very recently, focus has been to investigate ODE solvers for specific problem sets [17][18][19][20].The MATLAB software is widely used in engineering studies. Its ODE suite has seven powerful solvers, three of which are explicit schemes while the other four are implicit schemes. The ode45, ode23 and ode113 are the three explicit schemes while ode23t, ode23s, ode15s and ode23tb are the four implicit solvers. Interested readers are referred to the literatures [21-23] for mathematical details of the different solvers in the MATLAB ODE suite.Runge-Kutta methods are widely discussed in the literature [24][25][26][27]. The ode23 and ode45 solvers implement "Runge-Kutta formula of order 2 and 3" [21] and the "Dormand-Prince (4,5) pair"...

show abstract

An efficient Runge-Kutta (4,5) pair

Cited by 87 publications

References 23 publications

Free Convection Fluid Flow from a Spinning Sphere with TemperatureDependent Physical Properties

Free Convection Fluid Flow from a Spinning Sphere with TemperatureDependent Physical Properties

Functionally fitted explicit pseudo two-step Runge–Kutta methods

Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model

Contact Info

Product

Resources

About